SB    17 


Outlines 

For 

Methods  In  Arithmetic 


MANCHESTER 


OUTLINES  FOR  METHODS  IN 


ARITHMETIC 


RAYMOND  E.  MANCHESTER 

Head  Dept.  Mathematics,  Kent  State  Normal  School 


AUTHOR— The  Teaching  of  Mathematics 
Brief  Course  in  Algebra 
John  Citizen  and  His  School 
Teaching  Outlines,  etc. 


DERRICK  PUBLISHING  COMPANY,  PRINTERS 
Oil  City,  Pennsylvania 


\ 


COPYRIGHT  1922 
R.  E.  MANCHESTER 


CONTENTS 

.•» 

PART  I. 
General  Outlines. 

Introduction. 

Course  of  Study. 

Lesson  Types. 

Devices,  Games,  Drill  Charts,  Tests,  Marking  and  Diagnosis 

Problem  Analysis. 

PART  II. 
Subject  Matter  Outlines. 

Fundamental  Number  Ideas. 

Number  Appreciation  (Sense  Training). 

Language  (Number  System,  etc.) 

The  Fact  Groups. 

Addition  (Whole  Numbers). 

Subtraction  (Whole  Numbers). 

Multiplication  (Whole  Numbers). 

Division  (Whole  Numbers). 

Introduction  to  Fractions. 

Equivalent  Forms  (Fractions). 

Addition  (Fractions). 

Subtraction  (Fractions). 

Multiplication  (Fractions). 

Division  (Fractions). 

The  Three  Problems  in  Fractions. 

Decimal  Fractions. 

Percentage. 

Special  Topics. 


491083 


INTRODUCTION 

As  an  introduction  to  the  course  in  methods  students  are  asked  to 
consider  the  historical  background  of  arithmetic,  the  values  of  the  sub- 
ject and  some  of  the  modern  aims  and  purposes.  The  following  out- 
lines are  arranged  merely  to  serve  as  outlines  for  study. 


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1  HISTORICAL :  BACKGROUND 


Although^lUiS/tiot:ne€egsary;  tor  teachers  of  arithmetic  to  know 
the  complete"  history  of  the  subject  it  is  desirable  that  they  know  that 
arithmetic  is  one  of  the  subjects  of  the  common  people  and  to  know 
enough  of  its  beginnings  and  development  to  appreciate  its  importance 
as  a  subject  for  study.  In  a  course  of  this  kind  it  is  desirable  to  pre- 
sent the  outstanding  points  connected  with  the  rise  of  arithmetic  and 
the  needs  all  people  of  all  times  have  had  for  it.  It  may  inspire  some 
teacher  to  know  that  she  has  something  to  give  that  has  the  justifica- 
tion of  an  historical  background. 


OUTLINE  FOR  STUDY 

1.  Arithmetic  developed  among  trading  nations. 

2.  It  has  flourished  when  nations  have  been  active. 

3.  It  is  a  subject  that  has  been  kept  alive  through  the  interest  of 
the  common  people. 

4.  It  has  been  changed  to  meet  the  needs  of  the  common  people, 

5.  It  has  always  been  a  subject  for  study.     (Called  one  of  the  com- 
mon branches.) 

6.  It  is  now  changing  to  meet  new  demands  of  society. 

7.  Teachers  need  to  appreciate  the  necessity  of  meeting  new  needs. 


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VALUES  OK  ^ 


Most  of  us  accept  arithmetic;jw]thou't'  <}uestio^  as  a-  necessary  sub- 
ject in  the  curriculum.  This  is  bt^ca;u^ej  hcarly'^J^ti^tal  experiences 
are  connected  in  part  with  ideas  of  quantity.  Arithmetic  provides  a 
way  to  express  thoughts  about  quantity  and  its  measurement.  It  pro- 
vides other  things  as  well,  such  as  training  in  useful  habits  in  thinking, 
useful  habits  in  oral  and  written  expression,  a  group  of  facts  useful  in 
daily  contact  with  common  business  practice,  a  part  of  oral  and  written 
language  needed  in  ordinary  conversation,  an  introduction  to  advanced 
study  in  other  lines  of  mathematics  and  sciences,  a  group  of  facts  need- 
ed in  industry,  etc.  It  is  not  only  unnecessary  but  quite  impossible  to 
list  all  the  values  of  the  subject,  but  it  is  possible  and  also  desirable  to 
classify  the  values  by  groups  and  to  discuss  the  relative  values  of  the 
groups. 

Those  who  enjoy  the  study  of  mathematics  and  those  engaged  in 
vocations  and  professions,  in  the  practice  of  which  considerable  knowl- 
edge rf  mathematics  is  needed,  would  rate  the  values  of  arithmetic 
much  higher  than  would  those  who  derive  less  pleasure  from  and  have 
less  need  for  the  subject.  It  is  then  reasonable  to  suppose  that  differ- 
ent groups  of  people  would  stress  these  values  differently.  It  is  good 
educational  theory  to  accept  this  fact  and  arrange  courses  of  study  ac- 
cordingly. 

OUTLINE  FOR  STUDY 

1.  Provides  an  addition  to  our  language  that  is  necessary  to  ex- 
press fundamental  ideas  of  quantity,  form  and  position. 

2.  Provides  training  needed  for  appreciation  of  number  relations. 

3.  Provides  a  group  of  usable  facts  about  number  relations  and 
measurements. 

4.  Provides  a  group  of  processes  through  the  use  of  which  new 
facts  may  be  discovered. 

5.  Provides  training  in  methods  of  thought. 

6.  Provides  special  discussion  of  the  common  applications  of  facts 
and  processes  to  daily  affairs  of  life. 

7.  Provides  mental  pleasure. 

8.  Has  traditional  cultural  values. 

9.  Provides  a  valuable  group  of  mental  and  physical  habits. 


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••  i}    MODERN-  PURPOSES 

The  preser/t  nee£s,  of  .soeiety  for  any  subject  are  far  different  from 
the  needs  of  e:vt$t  a^w^a'rk'ajgq/  „  Every  teacher  knows  that  the  per- 
fection of  modern  business  methods  and  the  invention  of  machinery 
have  made  unnecessary  a  complete  knowledge  of  many  of  the  processes 
and  special  topical  discussions  and  have  created  needs  for  new  and  dif- 
ferent things.  The  teacher  of  arithmetic  must  be  prepared  to  adapt 
herself  to  changing  conditions  and  to  select  and  stress  those  parts  of 
the  subject  most  useful  to  those  who  pass  out  from  her  instruction. 

OUTLINE  FOR  STUDY 

1.  A  selection  of  facts  on  a  basis  of  utility  values 

2.  A  discussion  of  the  processes  with  stress  on  social  needs  for 
them. 

3.  A  selection  of  applications  based  on  the  organization  of  modern 
business. 

4.  A  discussion  of  the  logical  order  of  topics  with  reference  to  con- 
tinued study  of  mathematics. 

5.  A  discussion  of  mental  development  values. 

6.  A  discussion  of  the  study  of  arithmetic  from  the  point  of  view 
of  the  application. 


THE  COURSE  OF  STUDY 

When  considering  a  course  of  study  the  student  should  have  in 
mind  the  educational  theory  upon  which  the  course  is  based,  the  general 
plan  followed  in  making  the  course,  the  plan  of  organization  and  ar- 
rangement, the  purpose  of  the  course  and  the  selection,  classification 
and  distribution  by  grades  of  the  subject  matter.  The  following  out- 
lines are  arranged  to  guide  the  student  in  this  discussion. 


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EDUCATIONAL  THEORY 

1.  Education  defined  as  a  process  of  acquiring  knowledge. 

Under  such  a  theory  the  stress  is  upon  the  acquiring  of  facts  and 
skills.  It  is  a  definition  dominating  many  research  schools  as  well  as 
many  of  the  technical  and  vocational  schools.  The  effect  of  such  a 
theory  on  a  course  of  study  in  arithmetic  is  to  make  prominent  the 
memorizing  of  facts  and  the  development  of  speed  and  accuracy  in  the 
processes.  Less  attention  is  given  to  a  study  of  the  reasons  underlying 
the  steps  in  the  process  and  more  attention  is  given  the  learning  of  the 
steps  themselves. 

2.  Education  defined  as  a  development  of  the  mind. 

This  definition  of  education  stresses  the  development  of  the  mental 
processes  and  makes  the  power  to  think  the  goal.  Such  a  definition 
affects  a  course  of  study  by  placing  stress  on  the  thought  processes  in- 
volved in  a  study  of  number  relation.  Less  importance  is  given  to 
memorizing  of  facts  and  to  speed  and  accuracy  in  operation  and  more 
importance  is  given  to  analysis,  synthesis,  and  association  of  ideas. 
The  general  effect  upon  the  course  of  study  is  to  delay  the  opening  up 
of  topics,  to  give  more  time  to  their  development  and  to  keep  a  logical 
development  of  thought  intact. 

3.  Education  defined  as  adaptabilty. 

Many  feel  that  the  end  of  education  is  to  provide  one  with  enough 
general  information  to  enable  him  to  adapt  himself  to  any  situation. 
Exact  and  definite  instruction  is  not  given  so  much  prominence  as  the 
classification  of  knowledge  in  usable  shape.  In  arithmetic  the  effect 
is  to  stress  the  acquiring  of  general  principles  and  the  ability  to  find 
and  use  needed  facts  and  process  when  needed.  Less  importance  is 
given  the  knowing  of  facts  and  speed  and  accuracy  in  the  operations  and 
more  importance  is  given  the  use  of  reference  books  and  tables.  Analy- 
sis is  stressed  since  it  serves  adaptability. 

4.  Education  defined  as  power  to  serve  society. 

This  definition  is  based  on  the  consideration  of  an  individual  as  a 
unit  in  the  social  group.  It  suggests  the  perfect  individual  as  one  who 
can  serve  the  group  best.  A  course  of  study  based  upon  this  definition 
stresses  the  development  of  initiative,  inventiveness,  and  leadership. 
The  effect  on  the  course  in  arithmetic  is  to  place  less  attention  on  the 
facts  and  processes  for  their  own  sake  and  more  upon  the  use  of  them 
in  the  general  social  affairs  of  life. 

5.  Education  defined  as  a  preparation  for  immediate  needs. 

Such  a  definition  emphasizes  the  needs  of  the  present  rather  than 
of  the  future.  In  place  of  a  storing-up  of  knowledge  or  of  power  for 
future  unknown  situations  the  definition  would  suggest  a  solution  of 
each  problem  as  it  arises.  The  effect  on  arithmetic  is  to  eliminate 
greatly  the  time  given  the  subject  in  the  grades  and  to  cut  down  greatly 
the  subject  matter  of  facts  and  processes.  All  special  discussions 
would  be  left  until  such  time  as  they  may  enter  the  life  of  the  individual. 


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GENERAL  PLAN 

1.  A  Topical  Development. 

The  older  courses  were  arranged  to  develop  each  topic  rather  com- 
pletely before  a  new  one  was  opened.  This  plan  is  successful  only  when 
all  pupils  complete  the  course  and  is  not  followed  to  any  great  extent. 

2.  A  Spiral  Development. 

This  plan  calls  for  the  continuous  development  of  many  topics  in 
each  grade.  Many  topics  are  carried  forward  simultaneously.  This 
plan  is  not  used  in  pure  form  to  any  great  extent.  The  spiral  idea  is 
used,  however,  in  most  courses. 

3.  A  Combination  Plan. 

Most  courses  are  built  on  a  merged  plan  of  topical  and  spiral  devel- 
opment. 

4.  Development  on  Basis  of  Pupils'  Needs. 

This  plan  is  one  of  the  recently  suggested  ones  and  has  not  been 
sufficiently  tried  to  prove  its  worth. 


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ORGANIZATION 

I     By  Large  Units. 

1.  Primary  grades  1,  2,  3. 
Intermediate  grades  4,  5,  6. 
Junior  High  School  7,  8,  9. 
Senior  High  School  10,  11,  12. 

2.  Primary  grades  1,  2,  3,  4. 
Grammar  grades  5,  6,  7,  8. 
High  School  9,  10,  11,  12. 

3.  Primary  grades  1.  2,  3. 
Intermediate  grades  4,  5,  6. 
Grammar  grades  7,  8. 
High  School  9,  10,  11,  12. 

4.  Elementary  grades  1,  2,  3,  4,  5,  6. 
Junior  High  School  7,  8,  9. 
Senior  High  School  10,  11,  12. 

II    By  Small  Units. 

1.  By  years. 

2.  By  half  years. 

3.  By  months. 

4.  By  weeks  and  days. 


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PURPOSE  OF  COURSE 
I.     A  course  to  be  followed  in  detail  by  the  teacher. 

1.  A  manual  of  practice. 

2.  Definite  subject  matter  outlined. 

3.  Definite  method  suggested. 

4.  Definite  tests  and  examinations  provided  for. 

5.  Initiative  on  part  of  teacher  limited. 

6.  Fixed  possibility  for  success  of  teacher's  work. 

This  plan  is  a  valuable  one  for  inexperienced  teachers  and  for  use 
in  a  closely  organized  system  of  schools.  The  work  of  teachers  and 
pupils  is  standardized. 

II.     As  a  suggestive  outline. 

1.  Subject  matter  suggested  but  not  fixed. 

2.  Method  suggested  but  not  fixed. 

3.  Tests  and  examinations  suggested. 

4.  Possibilities  not  limited  but  directed. 

5.  Initiative  suggested. 

This  plan  is  a  valuable  one  for  experienced  teachers  who  have 
power  for  initiative  and  a  desire  to  do  excellent  teaching.  It  is  not 
usable  in  a  system  of  schools  so  organized  that  standardization  is  neces- 
sary. 

III.     To  establish  limits  for  work  by  grades. 

1.  Subject  matter  fixed  in  large  units. 

2.  Selection  within  units  in  hands  of  teacher. 

3.  Method  not  fixed. 

4.  Tests  and  examinations  fixed  with  reference  in  definite  time 
requirements  but  not  as  to  methods. 

5.  Possibilities  fixed  for  large  units  but  not  within  units. 

6.  Initiative  partially  limited. 

This  third  plan  is  useful  for  teachers  who  need  only  a  general 
guide  and  who  have  power  to  organize  the  work  within  limits.  A  good 
plan  for  use  in  the  smaller  systems  when  experienced  teachers  are  in 
charge. 


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SUBJECT  MATTER 
I.     Selection. 

(1)  For  later  use. 

(2)  For  organization  and  development  of  the  subject. 

(3)  To  serve  immediate  needs  of  the  pupil. 

(4)  For  mental  exercise. 

(5)  For  mental  recreation. 

(6)  To  prepare  for  other  subjects. 

II     Classification. 

1.  Fundamental  number  ideas. 

2.  Number  appreciation  (sense  training). 

3.  Development  of  language  (number  system). 

4.  Fact  groups. 

A.  Addition. 

B.  Subtraction. 

C.  Multiplication. 

D.  Division. 

E.  Measurement. 

5.  Processes. 

A.     Whole  numbers, 
a     Addition, 
b     Subtraction, 
c     Multiplication, 
d     Division, 
e     Measurement. 

B     Fractions. 

a.  Changing  to  equivalent  forms 

b.  Addition. 

c.  Subtraction. 

d.  Multiplication. 

e.  Division. 

f.  Simplification. 

c.  Solution  of  equations 

d.  Substitution. 

f.     Use  of  the  formula. 

6.  Applications. 

a.  Measurement. 

b.  Buying  and  selling. 

c.  Construction. 

d.  Production. 

e.  Communication. 

f.  Transportation. 

g.  Money  and  credit. 

h.     Analysis  and  solution  of  problems. 

III.     Distribution. 

1.  By  departments. 

2.  By  grades. 

3.  By  topics. 


LESSON  TYPES 

Although  this  topic  is  discussed  fully  in  the  classes  in  pedagogy  it 
is  desirable  for  those  preparing  to  teach  arithmetic  to  consider  lesson 
types  with  reference  to  this  particular  subject. 


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Development  Lesson   (Inductive) 

Purpose: — To  aid  the  pupil  discover  facts,  principles,  etc.,  through 
his  own  efforts.  The  procedure  is  to  develop  the  general  truth  from 
particular  experiences. 

Development   Lesson    (Deductive) 

Purpose: — To  aid  the  pupil  discover  facts,  prniciples,  etc.,  through 
his  own  efforts.  The  procedure  is  to  develop  the  particular  truth  from 
the  general  truth. 

Exposition  Lesson 
Purpose: — To  present  facts,  principles,  etc.,  to  the  pupils. 

Study   Lesson 

Purpose: — To  help  the  pupil  study  and  acquire  facts,  principles, 
etc.,  through  independent  effort. 

Drill    Lesson 

Purpose: — To  fix  facts  in  mind  and  establish  habits  of  operation 
and  of  thought. 

Appreciation   Lesson 

Purpose:— To  stimulate  pupils  to  effort  by  inspiring  them  with  ex- 
ample, by  arousing  interest  through  observation  of  the  work  of  others, 
or  by  appealing  to  the  aesthetic  emotions. 

Recitation  Lesson 
Purpose: — To  follow  up  assigned  work  of  any  kind. 

Review  Lesson 

Purpose: — To  organize  the  material  presented  and  to  recall  facts, 
principles,  etc.,  to  mind. 

Test  Lesson 
Purpose: — To  test  pupils  for  ability,  performance  and  development. 


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OUTLINES 


Inductive   Development   Lesson 


I.     Purpose. 


1.  To  help  the  student  learn  facts,  principles,  etc.,  by  helping  him 
organize  judgments. 

2.  To  teach  the  process  of  inductive  reasoning. 

II.  Steps. 

1.  Preparation. 
Statement  of  the  problem. 

2.  Development  of  all  particular  facts,  etc. 
Discussions  of  these  facts. 
Development  of  the  general  truth. 

III.  Application  of  the  general  truth  to  problem  solution. 

All  work  cannot  be  developed  in  this  way  since  the  aim  of  some 
lessons  is  not  to  discover  new  truths.  It  is  also  true  that  all  pupils  have 
not  had  sufficient  experience  to  make  this  type  of  lesson  useful. 

IV.  Discussion  of  the  method  used. 


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Deductive  Development  Lesson 

I.  Purpose. 

1.  To  help  the  pupil  learn  facts,  principles,  etc.,  by  having  him  or- 
ganize his  own  judgments. 

2.  To  teach  the  pupil  the  method  of  deductive  reasoning. 

II.  Steps. 

1.  Statement  of  the  general  truth. 

2.  Statement  of  particular  truths. 

3.  Verification  of  particular  truths  through  experiment. 

III.  Application  of  the  particular  facts  in  the  solution  of  problems. 

IV.  Discussion  of  the  method  used. 


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Exposition  Lesson 

I.  Purpose. 

1.     To  present  facts,  principles,  etc.,  to  pupils. 

II.  Steps. 

1.  Preparation. 

a.  General — The  pupils  must  be  mentally  ready  to  attend  to 
the  presentation.     The  teacher  hiust  develop  a  desire  to 
learn. 

b.  Particular — This  preparation  includes  all  review  of  needed 
material  and  all  instruction  leading  up  to  the  lesson  itself. 
New  words  must  be  defined,  class-room  material  distrib- 
uted, etc. 

2.  Procedure. 

a.  The  idea  should  be  developed  through  a  series  of  logical 
steps. 

b.  Each  step  should  be  carefully  taught  before  the  next  is  at- 
tempted. 

c.  Summary — The  points  of  the  lesson  should  be  discussed 
and  organized. 

III.  Drill. 

After  the  lesson  has  been  taught  and  the  results  organized  there 
should  be  a  short  drill  to  fix  the  lesson  in  the  minds  of  the  pupils. 


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The  Study  Lesson 

I.  Purpose. 

a.  To  teach  the  pupils  to  work  independently. 

b.  Help   the   pupil  to  realize   the   importance  of  learning   to 
study. 

II.  Pupils  should 

1.  Have  a  definite  statement  of  the  problem  to  be  solved. 

2.  Have  all  data  needed. 

3.  Establish  a  method  of  procedure. 

4.  Record  results  accurately. 

5.  Develop  power  of  analysis,  concentration,  etc. 

III.  The  problem  of  securing  independent  work  on  the  part  of   the 
pupil  is  two-fold. 

1.  Teaching  the  pupils  to  appreciate  the  factors  involved  in  study 

2.  Developing  interest  and  attention. 

IV.  It  is  necessary  to  teach  the  pupil  how  to  study  and  to  furnish  mo- 
tive for  study. 

V.  Factors  that  encourage  study. 

1.  The  teacher. 

a.  Should  state  the  problem  clearly. 

b.  Should  aid  the  pupil  with  all  suggestions  necessary. 

c.  Should  furnish  an  incentive  for  studying  the  lesson. 

2.  The  recitation. 

a.  Should  be  a  test  of  the  work  of  the  study  period. 

b.  Should  stimulate  the  pupil  to  a  desire  to  recite. 

c.  Should  recognize  individual  differences  in  ability. 

3.  The  suggestion  of  individual  interests  and  life  purposes. 


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Drill  Lesson 

I.  Purpose. 

1.     To  fix  facts  of  the  steps  in  a  process  in  mind  by  repetition  and 
establish  habits  of  operation  and  of  thought. 

II.  Steps. 

1.  Giving  a  motive  for  repetition. 

2.  Repetition  with  attention  on  the  particular  fact. 

III.  Means  cf  holding  attention  during  drill. 

1.  Devices,  games,  contests,  etc. 

2.  Placing  a  time  limit  and  recording  performance. 

3.  Appealing  to  emulation. 

IV.  Development  of  accuracy  in  practice. 

V.  Drill  should  be  regular  and  organized. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmiiiiiiiM 

The  Appreciation  Lesson 

I.  Purpose:     To  please,  to  inspire  and  to  stimulate  interest 

II.  Steps. 

1.  A  careful  preparation. 

a.  General  statements  to  develop  interest  and  get  pupils  men- 
tally ready  to  appreciate  the  lesson. 

b.  Particular  statements  to  recall  facts  and  principles  needed 
to  appreciate  the  lesson. 

2.  Presentation  of  the  lesson. 

a.  By  lecture  (teacher). 

b.  By  lecture  (outside  person). 

c.  By  trips  to  places  of  interest. 

d.  Through  class  discussion  and  conversation. 

e.  Through  dramatization  of  application  of  processes. 

f.  By  use  of  pictures,  charts  and  other  material. 

g.  By  use  of  games. 

h.     By  reports  given  by  students, 
i.      By  development  of  a  project. 

3.  Discussion  of  lesson. 

III.  Discussion  of  method. 


minimi n nun i ninii niinii iiuiiiii i iiiiiiiiiiiiiiiiiiiiiiiiini iiiiiiiiiiiiiiiiiiiiiitiinii inn in 1111 

The  Recitation  Lesson 

I.  Purpose. 

1.  To  test  the  work  done  in  the  study  period. 

2.  To  test  the  ability  of  the  pupil. 

3.  To  test  for  facts. 

4.  To  test  the  ability  of  the  pupil  to  present  knowledge. 

5.  To  test  for  analysis  and  organization. 

II.  Forms  of  the  recitation. 

1.  Question  and  answer — Oral  and  Written. 

2.  Discussion  open  to  pupils. 

3.  Presentation  by  pupils. 

III.  Questions  and  Answers. 

1.  Good  questions. 

a.  Concise  and  definite.     They  should  arouse  thought  on  the 
part  of  the  pupil. 

b.  Should  follow  in  logical  order  and  should  suggest  an  answer 
in  child's  own  words. 

2.  Questions  should  be  given  to  all  members  of  the  class,  before 
naming  the  pupil  to  answer  it. 

3.  No  set  order  in  questioning  pupils  should  be  followed. 

IV.  Discussion. 

1.  Should  give  pupil  an  opportunity  to  stand  on  his  feet  and  speak. 

2.  Suggest  the  organization  of  subject  matter. 


iiiiiiimiiiiimiiiimiiiiiiiiiiimiiiiiiimiiiiiiiiiiiiiiiiiim 

The   Review   Lesson 

I.  Purpose. 

1.  To  give  an  opportunity  for  organization  of  larger  units  of  the 
subject  than  the  daily  recitations  afford. 

2.  To  test  the  mastery  of  the  subject. 

3.  To  give  a  broader  perspective  of  the  work. 

II.  Kinds. 

1.  Short  range  review,  as  one  topic. 

2.  Long  range  review,  as  the  work  of  a  term  or  year. 

III.  Forms. 

1.  Organizing  the  facts  learned  around  some  central  topic. 

2.  Giving  the  pupils  a  topic  to  discuss. 

3.  The  application  of  the  facts  learned  to  some  new  central  topic 
or  situation. 

4.  The  review  should  reveal  to  pupil  and  teacher. 

a.  Growth  of  the  formation  of  habits. 

b.  Things  that  need  be  clearly  understood. 

5.  Time  should  be  taken  after  each  review  to  note  carefully  the 
situation  as  outlined. 


iiiiiiiimimiiimiiiiiiiiiiiiiiiiiimimmiiiiiiiiimiimiiiiiiiiiiiimm 

Test  Lesson 

I.  Purpose: — To  test  pupils  for  performance,  ability  and  development. 

II.  Plan. 

1.  Cover  the  ground. 

2.  Provide  sufficient  time. 

3.  State  the  aims  clearly. 

4.  Provide  definite  plan  for  marking 

5.  Establish  fairness. 

Tests  should  be  arranged  in  such  a  way  that  pupils  feel  that  an 
opportunity  is  offered  for  expression. 


DEVICES,  GAMES,  ETC. 

The  following  outlines  are  suggested  for  use  in  discussion  of  de- 
vices, games,  drill  charts,  tests,  marking  systems  and  diagnosis  of  fail- 
ing students. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 

DEVICES 

I.  Needs  for 

1.  An  aid  in  presentation  of  new  work. 

2.  An  aid  in  drill. 

3.  Graphic  presentation. 

4.  Saves  teacher's  time. 

II.  Classification. 

1.  Blackboard  drawings. 

2.  Charts  to  hang  on  wall. 

3.  Cards  to  use  as  perception  cards. 

4.  Number  arrangements  on  adjustable  frame. 

5.  Graded  tests,  and  exercises. 

6.  Blocks,  cardboard  figures,  etc. 

7.  Measuring  units. 

8.  Pictures. 

9.  Pegs,  splints,  balls,  seat  work  cards,  etc. 

10.  Sand  box. 

11.  Level. 

12.  Store. 

13.  Bank  windows,  etc. 

14.  Business  papers. 

III.  Selection. 

1.  Provision  for  age  of  pupils. 

2.  Provision  for  rural  or  city  schools, 

3.  Provision  for  community  life. 

4.  Provision  for  possibilities  of  school  room. 

5.  Provision  for  mental  strength  of  pupils. 

IV.  Purpose  and  Values. 

1.  To  supplement  presentation  of  new  work. 

2.  To  supplement  drill. 

3.  To  provide  for  variety  of  appeal. 

4.  To  provide  for  motivation. 

5.  To  provide  for  relaxation. 

6.  To  provide  for  connection  with  life  interests. 

7.  To  provide  for  group  activity. 

8.  To  save  teacher's  time. 

9.  To  stimulate  initiative. 


111,11,11 Ill, Illllllllllllllllllllllllllllllllllllll Illlllllllllllllllllllll Illllllllllllllllllllllllll I Illlllllllllllllllllllllllllllllllllllltlllllt 

GAMES 

I.  Needs  for. 

1.  To  offer  variety  of  appeal. 

2.  To  bring  arithmetic  into  play. 

3.  To  provide  for  relaxation. 

II.  Classification. 

1.  Counting  games. 

2.  Games  to  fix  number  facts. 

3.  Games  to  stimulate  sense  training. 

4.  Games  to  develop  processes. 

5.  Games  to  suggest  applications. 

III.  Selection. 

1.  Provision  for  age  of  pupils. 

2.  Provision  for  rural  or  city  school. 

3.  Provision  for  community  interests. 

4.  Provision  for  possibilities  of  school  room. 

5.  Provision  for  materials  available. 

IV.  Purpose. 

1.  To  supplement  presentation  of  new  work. 

2.  To  supplement  drill. 

3.  To  develop  appreciation  for  number  ideas. 

4.  For  individual  students. 

5.  For  group  work. 

V.  Values. 

1.  Variety  of  appeal. 

2.  Relaxation. 

3.  Connection  with  life  interests  of  pupils. 

4.  Stimulates  initiative  and  leadership. 

NOTE — See  list  of  games  following  this  outline. 


uimiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiM 

ARITHMETIC  GAMES 

1.     COUNTING. 

1.  There  are  many  games  used  in  the  very  early  stages  of  count- 
ing, but  the  most  commonly  used  one  is,  finger  play.     For  example,  a 
few  of  these  finger  plays  are: 

A.  Here  is  the  bee  hive. 
Where  are  the  Bees? 
Hidden  away, 
Where  nobody  sees. 
Soon  they  come  creeping 
Out  of  the  hive, 

One,  two,  three,  four,  five. 

In  this  game  the  closed  hands  is  the  bee-hive. 

B.  Five  little  children  sliding  on  the  floor 

One  tumbled  down  and  then  there  were  four. 

Four  little  children  laughing  with  glee 

One  tumbled  down  and  then  there  were  three. 

Three  little  children  sliding  toward  you 

One  tumbled  down  and  then  there  were  two. 

Two  little  children  sliding  for  fun 

One  tumbled  down  and  then  there  was  one. 

One  little  child  sliding  all  alone 

He  tumbled  down  and  then  there  were  none. 

Children  should  use  their  fingers  to  represent  the  number  of  chil- 
dren. 

C.  One,  two,  three,  four,  five, 
I  caught  a  hare  alive. 

Six,  seven,  eight,  nine,  ten, 
I  let  him  go  again. 

2.  Another  commonly  used  game  in  the  early  stages  of  counting  is 
the  following.: 

The  children  are  asked  to  skip  a  certain  number  of  times,  and 
to  count  each  skip.  The  same  method  may  be  used  with  hopping, 
jumping,  clapping  hands,  or  tapping  on  desks. 

3.  A  very  simple,  but  beneficial  game  for  the  children  in  their 
counting  is  as  follows: 

The  teacher  taps  oh  board  or  table  or  floor  with  pointer.  The 
children  listen  to  the  taps  and  count  them. 

4.  Number  touch  is  a  very  important  factor  in  the  early  training 
of  the  child. 

The  teacher  may  ask  a  child  to  close  his  eyes,  then  touch  his  hand 
a  certain  number  of  times,  and  have  him  state  the  number.  Probably, 
to  make  this  more  interesting,  the  children  could  try  this  with  one  an- 
other. 

5.  Another  very  beneficial  game,  used  to  improve  the  child's  count- 
ing is  to  play  the  blackboard  is  the  sky.    The  children  draw  stars  on  it, 
playing  it  is  just  twilight,  and  the  stars  are  just  beginning  to  show. 
The  children  may  count  silently,  and  then  in  unison,  telling  how  many 
there  are. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiim 

6.  The  teacher  may  have  envelopes  containing  various  pieces  of 
colored  paper  or  the  like.     She  writes  on  the  outside  (in  words)  the 
number  of  pieces  they  contain.    It  is  impossible  for  a  first  grader  to  read 
this,  but  the  teacher  herself  knows  how  many  pieces  are  in  the  en- 
velope.    She  passes  the  envelopes  and  the  children  count  and  tell  her 
the  number.     They  may  exchange  envelopes,  so  giving  each  child  a 
chance  to  improve  his  counting. 

7.  A  game  to  be  used  in  first  grade  is  one  where  the  children  string 
articles  such  as  corn,  wooden  beads,  etc.,  and  use  them  for  counting. 

8.  Have  two  or  more  children  extend  their  arms  at  the  same  time 
raising  one  or  more  fingers.     Have  all  the  children  guess  the  number 
of  fingers  raised  and  see  whose  guess  is  the  nearest  correct.     The 
teacher  keeps  score. 

9.  A  certain  number  as  "5"  is  chosen  for  "buzz."    Have  the  chil- 
dren count  either  1,  2,  3,  4,  5,  6,  etc.,  or  by  2's,  3's,  etc.,  but  each  number 
containing  5  or  its  multiple  is  omitted  and  the  word  "buzz"  used  in  its 
place.    This  is  good  training  for  multiplication  drills. 

II.     READING  AND  WRITING 

1.  Place  several  numbers  in  different  places  on  board,  point  to 
them,  and  have  children  read  them.    Individual  work  is  of  more  value. 

2.  Write  names  "four,"  "seven,"  etc.,  on  the  board.    Have  children 
copy  names  and  write  figures  corresponding  to  the  names. 

3.  Perception  Cards — Use  number  group  cards.    Hold  one  card  at 
a  time  in  front  of  the  children  just  for  an  instant.    The  one  saying  the 
number  first  gets  the  card.    When  all  cards  have  been  used  announce 
as  "winner"  the  one  holding  the  greatest  number  of  cards. 

4.  Place  a  box  upon  a  table  and  have  one  of  the  pupils  count  the 
objects  in  it  writing  their  number  on  the  board.    The  number  of  objects 
should  be  changed  as  each  pupil  counts. 

5.  Write  the  names  of  several  children  on  the  board  and  let  the 
children  count  the  letters.    This  can  be  done  in  groups  until  the  names 
of  all  the  pupils  have  been  used. 

6.  Jack  Homer  Pie. 

Write  numbers  on  quite  a  number  of  small  cards  and  drop  them  all 
in  something  used  to  represent  a  "pie."  (A  box  or  basket  will  do.)  Then 
choose  a  pupil  to  recite,  "Little  Jack  Homer."  When  he  comes  to  the 
line  "He  put  in  his  thumb,"  let  the  pupil  put  his  hand  in  the  "pie"  and 

draw  out  a  number.    When  he  says,  "And  pulled  out  a ,"  say  the 

number  drawn.  This  can  be  used  in  groups  as  well.  After  the  children 
have  become  accustomed  to  the  game,  you  can  leave  out  the  poem  and 
only  repeat  the  line,  "He  put  in  his  thumb  and  pulled  out  a  (number). 

7.  Draw  several  concentric  circles  on  blackboard.     Divide  all  cir- 
cles into  equal  parts  by  diameters.     Number  each  space.     While  one 
child  (who  is  blindfolded)  is  letting  the  pointer  wander  over  the  circles, 
the  other  children  are  saying  in  concert  "Tic  tac  toe,  round  I  go;  hit  or 
miss,  I'll  take  this."    As  children  say  "I'll  take  this,"  the  child  is  un- 
blinded  and  reads  the  number  to  which  he  is  pointing.    If  his  pointer  is 
on  a  line  he  must  read  the  number  on  both  sides  of  the  line. 

8.  Card  Game. 

Cards  for  the  "Jack  Horner  Game"  are  used.  These  cards  are 
placed  in  a  basket.  One  child  must  stand  by  the  basket  and  pick  out 
the  numbers.  The  other  children  go  to  the  board.  A  card  is  drawn  by 
the  child  at  the  basket.  As  he  reads  it,  those  at  the  board  write  the 
number.  The  instructor  corrects  numbers  at  the  board. 


imiimiiiiiiiiiimiiiimiiiiiiiiiiimimiiiiiiiiiiiiiiiiiiiiiimiiiim 

III.     Many  of  the  following  drills  are  usable  for  addition  drill,  but  they 
may  be  used  in  subtraction,  multiplication  and  division,  too. 

1.  When  the  addition  facts  are  first  being  worked  out,  games  may 
be  useful.    Cards  made  to  represent  dominoes  are  useful.     Each  of  the 
children  is  furnished  with  a  card.    They  come  to  the  front  in  turn  and 
show  card  long  enough  for  other  children  to  see  it.     The  child  quickly 
puts  the  card  behind  him  and  asks  "How  many?" 

2.  Instructor  places  cards  in  chalk  tray  in  regular  order.    A  card, 
having  a  number  on  it,  as  6,  is  held  up  by  the  teacher.    Child  takes  two 
cards  from  tray  whose  sum  is  6.    This  must  be  continued  until  all  cards 
of  the  sum  6  are  found.     Another  number  is  then  used  and  the  game 
continues. 

3.  Combinations  without  answers  are  written  on  the  board.    Cards 
are  placed  in  ledge,  which  contain  single  numbers.    Teacher  points  to  a 
combination  of  numbers  and  children  go  to  cards  and  pick  out  answer 
on  the  cards.     Child  may  point  to  the  combination  on  the  board  after 
the  teacher  has  pointed  to  a  card. 

4.  Distribute  cards  on  which  are  different  numbers  to  the  children. 
Pick  out  a  number  such  as  5,  which  several  children  have  on  their 
cards.    Call  for  this  number  and  the  children  who  have  the  5  must  place 
it  on  the  board. 

5.  Single  number  perception  cards  are  used.    One  number  is  placed 
on  the  board.     A  number  placed  on  the  board  by  the  teacher  is  added 
to  the  number  which  the  child  has  on  his  card.     The  number  on  the 
board  is  changed.    If  child  cannot  add  this  the  class  is  called  on  to  re- 
spond. 

6.  Write  a  combination  on  the  board,  let  class  see  it  and  erase. 
Scores  for  correctness  may  be  kept  by  rows. 

7.  When  a  list  of  combinations  has  been  written  on  board  pupils 
in  turn  tell  the  answers,  then  one  child  alone  gives  all  the  answers. 

8.  Have  a  list  of  combinations  on  board.     Teacher  names  an  an- 
swer and  asks  a  pupil  to  point  to  the  possible  combinations  that  make 
this  answer. 

9.  A  picture  of  a  wagon  filled  with  hay  or  cakes  of  ice.    Each  piece 
has  a  written  number  combination  on  it.    If  one  child  can  name  all  the 
results  of  combinations  he  may  be  horse  to  draw  the  wagon. 

10.  Draw  a  picture  of  creek  with  stones  in  it.     Each  stone  has  a 
number  combination  written  on.  it.     If  a  child  fails  to  give  correct  re- 
sults he  has  fallen  in;  that  is,  he  has  been  unable  to  cross  without  get- 
ting the  correct  results. 

11.  Draw  a  picture  of  railroad  track.    Have  stations  all  along  the 
way  indicated  by  number  combinations.    If  a  child  can  give  correct  re- 
sults of  combinations  he  may  be  conductor  all  through  the  trip. 

12.  Make  a  list  of  different  combinations  as: 

87264 
43664     etc. 

Then  let  each  child  see  if  he  can  make  a  list  after  having  observed 
what  was  written. 

13.  Write   a   variety   of   division,   multiplication,    subtraction   and 
addition  combinations  on  the  board,  each     in  a  different  color.     Then 
give  the  children    each  a  different  color  of  chalk  and  let  each  one  see 
how  many  combinations  he  can  write.     The  one  that  gets  the  highest 
number  correct  is  the  winner. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiN 

14.  Let  one  child  stand  before  the  class  and  have  him  think  of  a 
certain  number.     Then  have  the  children  guess  by  asking  him  ques- 
tions.    Someone  may  say  4  plus  4.     If  this  answer  is  not  correct  they 
guess  until  someone  has  guessed  the  correct  answer,  then  the  winner 
is  the  next  one  to  stand  before  the  class. 

15.  The   teacher  should   place   a   number   of   combinations,   plus, 
minus,  times  and  divide,  on  the  board.    She  tells  one  child  to  close  his 
eyes,  then  she  sends  another  child  to  the  board  and  has  him  write  a 
combination  on  the  blackboard.     He  writes  20.     The  first  child  is  told 
to  open  his  eyes  and  guess  whether  20  means  10  plus  10,  or  10  times  2. 
If  he  does  not  guess  the  answer  correctly  the  first  time  he  does  not  get 
a  very  high  score,  but  if  he  guesses  10  plus  10,  and  that  is  what  the  boy 
was  thinking  about  when  he  wrote  it  on  the  board,  his  score  is  10. 

16.  Place  a  row  of  combinations  upon  the  board 

2         3         7         8         8         5         C 
2632424     etc. 

Two  children  are  sent  to  the  blackboard,  one  of  the  children  starting 
with  the  first  problem,  and  the  other  starting  with  the  last  and  work- 
ing toward  each  other.  When  the  teacher  says  "start"  each  child  en- 
deavors to  get  more  of  the  answers  in  a  certain  length  of  time  than  the 
other  one.  When  the  teacher  says  stop,  she  counts  and  sees  which  one 
has  the  highest  number  correct.  This  teaches  the  children  to  work 
rapidly.  Some  days  one  could  have  the  boys'  row  compete  with  the 
girls'  row,  one  row  with  another,  or  one  division  with  another. 

17.  Place  the  children  in  a  circle  and  give  each  one  a  certain  num- 
ber.   The  teacher  stands  outside  of  the  circle  and  says  "4  plus  4"  and 
as  she  says  this  number  she  throws  the  ball  into  the  circle.    The  child 
that  is  the  number  catches  the  ball  and  throws  the  ball  to  the  teacher. 
The  children  who  do  not  catch  the  ball  as  their  numbers  are  called, 
stand  in  the  circle  until  they  do  catch  the  ball. 

18.  Bird  Catcher — Children  may  be  arranged   in  a  circle  and  a 
number  card  given  to  each  one.     One  child  stands  in  the  center  and 
gives  problems,  the  answers  of  which  are  within  the  numbers  assigned. 
He  may  say,  "How  many  are  5  roses  and  3  roses?"    Then  the  person 
having  the  number  8  holds  it  up.    Thus  he  has  "caught  the  bird." 

19.  Number  Tug — Pupils  are  arranged  as  for  a  tug  of  war,  each 
side  having  a  captain.    Then  the  captain  of  one  side  gives  number  com- 
binations to  the  other  group,  such  as,  "3  times  3"  or  "3  times  4."    Each 
pupil  takes  his  seat  if  he  fails  to  give  correct  answer. 

20.  Circle  Addition — A  circle  of  this  sort  may  be  drawn  with  in- 
terior and  exterior  numbers.    The  class  may  be  divided  into  two  groups 
with  a  captain  for  each  side.    Each  captain  may  then  call  on  those  of 
the  opposite  side  to  give  the  sum  of  the  interior  and  exterior  numbers 
to  which  he  points.    Pupils  are  to  sit  down  when  they  fail  to  answer 
quickly.    The  game  may  be  used  for  multiplication,  division,  or  subtrac- 
tion by  changing  the  numbers  to  suit  the  process. 

21.  When  the  above  game  is  used  for  the  multiplication  process 
the  pupil  may  be  asked  to  add  a  given  number  to  the  product,  or  to  sub- 
tract a  given  number  from  the  product. 

22.  A  column  of  figures  are  placed  on  the  board.    The  pupils  may 
add  each  number  to  the  number  at  the  side.    This  game  may  be  varied 
by  having  the  child  begin  at  the  top  and  add  down  or  begin  at  the  bot- 
tom and  add  up.    This  game  may  also  be  used  for  subtraction  by  plac- 
ing a  larger  number~opposite  the  column;  for  example,  "15."    Then  it 
may  be  asked,  "This  number  and  how  many  are  15?" 


iiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiiimim 

23.  Hide  and  Seek — The  teacher  places  statements  on  the  board 
having  one  number  hidden;  such  as,  3  plus       ?  equals  7;   5  plus       ? 
equals  9;  ?    plus  4  equals  8.    Pupils  obtain  good  training  in  repetition 
of  number  forms  through  this  game. 

24.  Make  three  arches  of  paper.     Fasten  the  largest  arch  to  the 
floor  with  thumb  tacks.     Place  the  two  smaller  arches  on  either  side 
of  the  large  arch  and  fasten  with  thumb  tacks.    Children  stand  at  cer- 
tain distances  from  the  arches  and  take  turns  trying  to  roll  a  small  rub- 
ber ball  under  one  of  them.     Place  the  score  under  the  arch.     After 
there  have  been  two  turns  around  add  the  scores. 

25.  Have  an  equal  number  of  children  seated  in  each  row.     Give 
the  child  in  the  front  seat  of  each  row  a  bean  bag  or  an  eraser.     The 
children  hold  up  their  hands  ready  to  receive  the  object  when  the  sig- 
nal is  given.     The  children  pass  the  objects  back  over  their  heads. 
When  the  last  child  in  the  row  gets  the  bag  he  runs  quickly  to  the  chalk 
tray  and  lays  the  bag  in  it.     The  child  who  gets  there  first  wins  the 
game  for  his  row.    The  score  is  kept  for  each  row. 

26.  Hoop  Game — Have  a  hanging  hoop  in  which  there  is  a  bell. 
The  children  may  throw  bean  bags  through  this  hoop.    If  the  bag  hits 
the  bell  the  throw  counts  only  2.    Every  ball  that  goes  through  the  hoop 
without  ringing  the  bell  counts  10.    In  ten  throws  each,  two  pupils  may 
have  the  following  score: 

0         0         2       10         0         0       10         0 
10         0         0         0         2         0       10         2 

Find  the  score  of  each  and  tell  who  wins  the  game. 

27.  Odd  or  Even — This  is  a  very  old  game.    Two  persons  play  this 
game.    Each  takes  10  peas  or  marbles.    One  of  the  children  places  his 
hands  behind  his  back  and  arranges  the  objects  to  suit  himself.     He 
then  stretches  out  his  closed  hand  and  says,  "Odd  or  Even."     If  the 
other  child  guesses  correctly  he  receives  a  maible  and  if  incorrectly 
he  pays  one.    The  other  child  says  in  the  latter  case:     "Give  me  one  to 
make  it  odd  or  even  (as  the  case  may  be)." 

28.  The  teacher  writes  problems  upon  the  blackboard  in  rapid  suc- 
cessibn.    One  child  is  asked  to  stand  and  give  results  as  rapidly  as  pos- 
sible.   The  answers  are  not  written.    As  soon  as  one  child  makes  a  mis- 
take another  child  takes  his  place. 

29.  Thumbs  Up — One  pupil  acts  as  a  leader  in  this  game.     Each 
player  has  been  given  a  number.    He  sits  with  one  thumb  up.    The  lead- 
er says,  "Simon  says  15"  at  which  the  thumbs  of  3  and  5  (factors  of  15) 
must  be  turned  down.    If  Simon  says  12,  then  the  thumbs  2,  3,  4  and  6 
must  be  turned  down. 

30.  A  leader  is  chosen  to  give  each  child  a  number.    After  this  is 
done,  the  leader  names  any  number  below  any  prescribed  limit,  such  as 
25.    All  children  whose  given  number  is  a  factor  of  the  number  named 
by  leader  must  then  change  their  seats.    In  case  there  is  but  one  dis- 
tinct factor  for  the  number  named,  as  in  the  case  of  25,  the  pupil  whose 
number  is  that  one  single  factor  rises  and  bows.    If  a  child  fails  to  rise 
he  is  tagged. 

31.  This  game  may  be  used  as  a  drill  for  either  multiplication, 
division,  or  fractions.    Draw  a  2-foot  square,  which  is  then  to  be  divided 
into  144  smaller  squares.    The  products  of  the  tables  are  then  written 
promiscuously  in  these  columns,  and  the  teacher  points  to  two  num- 
bers, asking  the  pupils  to  divide,  multiply,  or  get  the  fractional  parts. 

32:  This  is  merely  a  simple  drill,  as:  4  times  4,  minus  2,  plus  4, 
times  4,  divided  by  3,  minus  6,  divided  by  3,  times  7,  plus  7,  divided  by 
7  equals  ? 


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DRILL  CHARTS 

I.  Purpose. 

1.  To  arrange  drill  on  definite  and  systematic  basis. 

2.  To  provide  teacher  with  a  labor  saving  device. 

3.  To  distribute  time  according  to  difficulty  of  work. 

II.  Plan. 

1.  To  provide  for  a  number  of  days  (perhaps  30). 

2.  To  select  subject  matter. 

To  arrange  subject  matter  in  order  of  difficulty. 
To  make  up  chart  so  that  all  work  is  provided  for. 
To  provide  for  motivation  of  work. 

III.  Classification. 

1.  Reading  numbers. 

2.  Counting. 

3.  Facts. 

a.  Addition 

b.  Subtraction 

c.  Multiplication 

d.  Division 

e.  Measurement 

4      Processes. 

a.  Addition 

b.  Subtraction 

,.     Multiplication 

d.  Division 

e.  Measurement 

5.  Fractional  equivalents. 

6.  Fractional  operations. 

7.  Decimal  equivalents. 

8.  Aliquot  parts. 

IV.  Value. 

1.  To  pupil. 

2.  To  class. 

3.  To  teacher. 


ADDITION    DRILL    CHART— 20  Days 


1 

2 

11 

12 

13 

14 

11 

12 

13 

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5 

7 

8 

9 

6 

5 

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— 

ADDITION    DRILL   CHART— (Continued) 


9 

10 

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13 

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6 

7 

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8 

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8 

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ADDITION    DRILL   CHART — (Continued) 


17 

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8 

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TESTS 

I.  .  Needs  for 

1.  To  furnish  information. 

2.  To  provide  incentive. 

3.  To  provide  basis  for  classification  of  students. 

4.  A  means  of  associating  ideas. 

II.  Purposes. 

1.  Indirect  purpose  (General  Information). 

2.  Direct  purpose  (To  test  for  particular  knowledge). 

III.  Classification. 

1.  General  information. 

2.  Performance,  Accuracy,  Speed. 

3.  Special  information. 

4.  To  test  for  development. 

IV.  Plan  in  giving  a  test. 

1.  Cover  the  ground. 

2.  Provide  sufficient  time. 

3.  Clear  aim. 

4.  Provide  definite  plan  for  marking. 

5.  Establish  fairness. 

V.  Values. 

1.  To  the  pupil. 

2.  To  the  class  (as  a  basis  for  marking). 
:\.     An  aid  to  organization  of  work. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJIIIIIIM 

MARKING 

I.  Needs  for 

1.  To  provide  a  record  of  work  completed. 

2.  To  stimulate  pupils  to  effort. 

3.  To  provide  a  basis  for  classification  of  pupils. 

II.  Purposes 

1.  To  record  performance. 

2.  To  record  ability. 

3.  To  record  habits  of  study. 

4.  To  record  habits  of  accuracy. 

5.  To  record  habits  of  neatness. 

6.  To  record  attitude  toward  teacher,  work  and  class. 

III.  Classification 

1.  Daily  class  recitation. 

2.  Knowledge  of  topics. 

3.  Final  knowledge  of  subject. 

4.  Attitude  and  attempt. 

IV.  Plans 

1.  Five  point  scale. 

2.  One  hundred  point  scale. 

3.  Combination  scales. 

4.  Two  point  scale  (all  or  nothing). 

V.  Values 

1.  Temporary  record. 

2.  Permanent  record. 

3.  Motivation. 

4.  Basis  for  classification  of  pupils. 


iiiiiiiniiniiiiiiiiiiiiMiiiMiiiiiiiiiiiiiiiiniiiiniiiinHiiniiiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiniiiiiiiiiiiiiiMiMniiiin iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii 

DIAGNOSIS 

I.  Needs  for 

1.  To  provide  basis  for  individual  instruction. 

2.  To  properly  grade  pupils. 

0.  To  determine  possibilities  of  pupils. 

II.  Purposes  t 

1.  To  keep  pupils  advancing  as  rapidly  as  possible. 

2.  To  protect  individuals  from  class  averages. 

3.  To  aid  teacher  in  classification  of  pupils. 

III.  Plan 

1.  To  arrange  subject  matter  in  order  of  difficulty. 

2.  To  provide  charts  for  recording  the  results  of  testing. 

3.  To  provide  careful  tests. 

4.  To  record  results. 

IV.  Values 

1.  To  pupil. 

2.  To  class. 

3.  To  teacher. 


111 iiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim 

PROBLEM  ANALYSIS 

First — Read  the  problem. 

Second — State  what  is  to  be  found. 

Third — State  what  is  given. 

Fourth — State  the  general  method  of  Solution. 

Fifth — State  the  steps  in  the  solution. 

Sixth— State  the  results  of  these  operations. 

Seventh — State  the  answer. 

1.  A  man  sells  a  farm  for  $4,300.    The  commission  of  the  salesman 
is  5%.    The  deed  costs  $2.00,  the  abstract  $7.00,  and  the  incidental  ex- 
pense amounts  to  $27.78.    Find  the  net  amount  the  man  receives. 

2.  I  am  to  find  the  net  amount  due  the  man  who  sells  the  farm. 

3.  I  know  the  selling  price  is  $4,300.    I  know  the  salesman  is  to  re- 
ceive a  commission  of  5%.     I  also  know  that  the  other  expenses  are 
$2.00,  $7.00  and  $27.78. 

4.  The  general  method  is  to  subtract  the  sum  of  the  commission 
and  the  expenses  from  the  selling  price. 

5.  The  first  step  is  to  find  5%  of  $4,300.    The  next  step  is  to  add 
the  amounts,  $2.00,  $7.00  and  $27.78.     The  next  step  is  to  subtract  the 
sum  of  expense  from  $4,300. 

6.  5%  of  $4,300  is  $215.     The  sum  of  $215,  $2.00,  $7.00  and  $27.78 
is  $251.78;  $251.78  from  $4,300  equals  $4,048.22. 

7.  The  answer  is  $4,048.22. 


PART  II. 
Subject   Matter   Outlines 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiimiiiiiiiiiiiiiiiin 

FUNDAMENTAL  NUMBER  IDEAS 

All  number  processes  are  based  upon  a  few  very  siniple  prin- 
ciples. It  is  necessary  that  these  be  brought  to  the  attention  of  pupils 
very  early.  This  is  done  indirectly  These  principles  are  built  upon 
some  truths  connected  with  quantity,  form  and  position.  Although 
these  number  ideas  are  so  very  simple  that  we  are  always  inclined  to 
disregard  them  all  the  processes  are  based  on  them.  For  example  one 
of  the  simple  ideas  about  quantity  is  that  a  quantity  may  be  increased 
and  upon  this  idea  the  processes  of  addition  and  multiplication  are 
based. 

As  suggested  above  the  ideas  we  are  concerned  with  when  we 
build  up  the  science  of  arithmetic  are  those  connected  with  quantity, 
form,  and  position.  Of  these  three  groups  of  ideas  we  are  especially 
concerned  with  the  group  relating  to  quantity.  This  explains  why 
arithmetic  is  sometimes  spoken  of  as  a  subject  which  provides  an 
answer  to  the  questions — How  much?  or  How  many? 


OUTLINE  FOR  STUDY 

I.  Quantity 

1.  A  quantity  may  be 

a.  Increased 
(by  addition) 

(by  multiplication) 

b.  Decreased 
(by  subtraction) 

c.  Divided 
(by  division) 

2.  Two  or  more  like  quantities  may  be 

a.  Combined 
(by  addition) 

b.  Compared 

(by  finding  the  difference — subtraction) 
(by  measurement — division) 

II.  Form  (only  standard  forms  are  considered). 

1.  recognition 

2.  oral  names 

3      written  names 

4.  classification 

5.  measurement 

III.  Position  (very  little  attention  given  to  this  topic). 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH^ 

NUMBER  APPRECIATION 

Although  no  definite  attempt  is  made  to  give  general  sense  train- 
ing it  is  desirable  to  give  drills  and  presentations  necessary  to  enable 
pupils  to  appreciate  the  number  aspects  of  those  things  they  see,  hear 
and  touch.  This  is  not  at  all  difficult  and  takes  up  very  little  time. 
The  use  of  cards,  pictures  and  other  concrete  material  Is  suggested. 

OUTLINE  FOR  STUDY 

I.  Vision 

1.  Number  groups 

2.  Recognition  of  measuring  units 

3      Estimation  of  value  (length,  weight,  etc.) 
4.     Recognition  of  standard  forms 

II.  Hearing 

1.  Continuity  of  sound 

2.  Intensity  of  sound 

3.  Groups  of  sounds 

III.  Touch 

1.  Number  groups 

2.  Recognition  of  standard  forms 

3.  Estimating  weights 


LANGUAGE 


A  part  of  the  work  in  teaching  arithmetic  is  devoted  to  the  neces- 
sary addition  to  our  language  to  make  possible  the  expression  of  num- 
ber ideas.  Not  only  must  pupils  learn  to  appreciate  number  truths 
but  they  must  learn  to  express  such  ideas  through  speech  and  in  writ- 
ing. Again  they  must  learn  to  understand  the  expression  of  others. 

The  most  important  part  of  this  work  is  that  connected  with  what 
we  refer  to  as  the  Arabic  number  system.  The  Arabic  number  system 
is  a  part  of  our  language.  When  teaching  pupils  the  number  system 
all  principles  of  language  teaching  must  be  used. 

Another  part  of  the  language  instruction  in  arithmetic  is  that 
given  over  to  proper  sentence  constructions  when  number  thoughts 
are  to  be  expressed.  Questions  connected  with  a  choice  between  two 
and  one  are  three  or  two  and  one  is  three  are  questions  of  language. 

In  the  processes  there  are  both  oral  and  written  forms  to  consider. 
Throughout  all  work  in  arithmetic  children  should  be  taught  to  make 
correct  and  complete  statements.  The  use  of  language,  both  oral  and 
written,  is  to  give  us  an  instrument  to  express  our  thoughts  one  to 
another.  Much  of  the  difficulty  in  arithmetic  may  be  traced  to  care- 
less statements.  Exact  statements  should  be  made  not  only  in  the 
solution  of  a  problem  but  also  in  the  reading  of  a  problem.  Often 
children  do  not  understand  what  is  asked  for  when  a  problem  is  given 
for  solution.  An  answer  in  a  problem  solution  must  of  necessity  be 
correct  or  incorrect.  To  arrive  at  a  correect  solution,  one  must  read 
the  problem  correctly  and  then  speak  or  write  correctly. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 

THE  LANGUAGE  DEVELOPMENT 

One  phase  of  the  study  of  arithmetic  is  that  connected  with  the 
development  of  the  necessary  language  to  express  number  thoughts. 
The  mastery  of  the  Arabic  symbol  system  constitutes  the  main  part  of 
this  addition  to  the  general  fund  of  language. 

OUTLINE  FOR  STUDY 

A.     The  number  system 

a.  Base  symbols     (0123456789) 

(1)  Recognition 

(2)  Oral  names 

(3)  Written  forms 

(4)  Order 

(5)  Relative  values 

b.  Combination  symbols    (10-11-etc.) 

1.     Place  value 

2      Decimal  (ten)  law  in  classification 

(a)  2  digit  numbers. 

(b)  3  digit  numbers. 

(c)  4  digit  numbers, 
etc. 

3.  Recognition 

4.  Oral  names 

5..    Written  forms 

6.  Order 

7.  Relative  Values 

(Note) — The  above  outline  is  to  be  followed  in  a  general  way 
when  studying  Roman  numbers. 

B      Sentence  arrangement 

(a)  Choice  of  words 

(b)  Structure  of  sentences 

C.  Written  forms  for  the  processes. 

(a)  Whole  numbers. 

(b)  Fractions. 

D.  Oral  forms  for  the  processes. 

(a)  Whole  numbers 

(b)  Fractions. 

E.  Special  language  forms. 

This  includes  those  language  forms  that  do  not  classify  in  the 
above  outlines. 


THE  FACT  GROUPS 

There  are  certain  groups  of  facts  the  teacher  must  present.  It 
is  convenient  for  the  teacher  to  have  these  groups  arranged  in  the 
order  of  difficulty  and  in  such  form  as  to  be  accessible  when  needed. 
The  common  fact  groups  are  those  of  addition,  substraction,  multipli- 
cation, division  and  measurement. 

In  the  following  classification  of  the  fact  groups  the  arrangement 
suggested  is  based  on  the  results  of  systematic  study  of  work  done  in 
primary  grades.  The  outlines  are  meant  to  be  suggestive  and  they 
may  be  changed  to  meet  the  needs  of  special  classes. 

The  addition  and  subtraction  groups  are  arranged  in  groups  on 
a  basis  of  difficulty  and  are  to  be  taught  together. 

The  multiplication  facts  and  division  facts  are  arranged  with  ref- 
erence to  products.  They  are  also  grouped  according  to  difficulty. 

In  the  outline  covering  the  measurement  facts  the  suggestion  is 
that  the  work  should  progress  by  grades  with  regard  for  the  pupils 
needs  and  experience. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiinii 111 iitiiiiiiiiiiiiiiiimiiiiiiMiiiiiiiiiiiiiiiiiiminiiiiiiiiiiiiimiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiinii iiinn 

ADDITION  FACTS 


(1)          111111111 
123456789 


(2)  2     2     3     *4     5 

32345 


(3)        222226678 
456784678 


(4)      23333334444 
94567895789 


(5)  000000000 

123456789 


(6)        5555666778 
6789789899 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIN 

SUBTRACTION  FACTS 


(1)  233445566778899     10     10 

112131415161718       1       9 


(2)  5  5  4  6  8  10 

32234  5 


(3)  6     6     7     7     8     8     9     9     10     10     10     10     12     14     16     18 

24252627       2       86       4       6       7       8       9 


(4)  11     11     7     7     8     8     9     9     10     10     11     11     12     12     9     9 

2       9343536       3       7       3       8       3       945 

11     11     12     12     13     13 
474849 


(5)  1234567 

0000000 


(6)   11     11     12     12     13     13     14     14     13     13     14     14     15     15     15     15 
5657585967686978 

16     16     17     17 
7989 


MULTIPLICATION  AND  DIVISION  FACTS 


CHART  I 

Group  1 

4 

6 

8 

9                   10 

15 

2     2s 

2     3s 

2     4s 

3     3s             2     5s 

3     5s 

3     2s 

4     2s 

5     2s 

5     3s 

Group  2 

12           14           16  18 

2  6s  2  7s         4  4s  2  9s 
6  2s  7  2s        2  8s  9  2s 

3  4s                    8  2s  3  6s 

4  3s  6  3s 

Group  3 


20 

25 

30 

35 

40 

45 

2 

10s 

5 

5s 

3 

10s 

7 

5s 

4 

10s 

9  5s 

10 

2s 

10 

3s 

5 

7s 

10 

4s 

5  l.i 

4 

5s 

5 

6s 

8 

5s 

6 

4s 

6 

5s 

5 

8s 

50 

60 

70 

80 

90 

100 

5 

10s 

6 

10s 

7 

10s 

8 

10s 

9 

10s 

10  10s 

10 

5s 

10   6s 

10 

7s. 

10 

8s 

10 

9s 

Group 

4 

21 

24 

27 

28 

32 

36 

3 

7s 

3 

8s 

9 

3s 

4 

7s 

4 

1 

is 

4  9s 

7 

3s 

8 

3s 

3 

9s 

7 

4s 

8 

4s 

9  4s 

4 

6s 

6  6s 

6 

4s 

Group  5 

42 

48     49     54 

56 

63 

64     72 

81 

6  7s 

6  8s    7  7s    6  9s 

7  8s 

7  9s 

8  8s    9  8s 

9  9s 

7  6s 

8  6s          9  6s 

8  7s 

9  7s 

8  9s 

CHART  II 

Group  1 

22 

24      33 

36 

44 

48      55 

60 

2  11s 

2  12s    3  11s    3 

12s    4 

11s    4 

12s    5  lls 

5  12s 

11  2s 

12  2s   11  3s   12 

3s   11 

4s   12 

4s   11  5s 

12  5s 

Group  2 

66 

72       77 

84 

88 

96 

99 

6  11s 

6  12s     7  11s 

7  12s 

8  lls 

8  12s 

9  lls 

11  6s 

12  6s    11  7s 

12  7s 

11  8s 

12  8s 

11  9s 

Group  3 

108        110        120        121        132        144 

9  12s     10  11s     12  10s  11  11s     11  12s     12  12s 
12  9s     11  10s     10  12s  12  11s 


iHHumuiHHirHiiiiiiiiiimiiiiiiiiiiNimiiiiiiiimiiiimiiiiiiiiiiiiiiim 

MEASUREMENT  UNITS 

There  is  a  distinction  made  between  a  knowledge  of  the  measuring 
units  with  their  relationships  and  the  operation  of  measuring. 

The  following  list  shows  a  suggestive  grouping  of  the  measuring 
units  with  their  relationships  by  grades.  The  work  in  measuring  should 
follow  the  presentation  of  units  and  facts  very  closely. 

GRADE  I 

Cent  5  cents  1  nickel 

dime  10  cents  1  dime 

5  cent  piece  two  nickels  1  dime 

pint  2  pints  1  quart 

quart 

inch 

foot  12  inches  1  foot 

GRADE  II 

Quarter  25  cents  equals  1  quarter 

Half  Dollar  5  nickels  equals  1  quarter 

Dollar 

Minute  as  used  in  conversation 

Hour 

Day 

Week 

GRADE  III 

ounce  16  ounces  equals  1  Ib. 

pound 

dozen  12  equals  1  doz. 

degrees  of  temperature  To  read  thermometer 

yard  3  ft.  equals  1  yd. 

36  in.  equals  1  yd. 

month  30  days  or  31  days  equals  1  month 

GRADE  IV 

gallon  4  qts.  equals  1  gal. 

peck  8  qts.  equals  1  peck, 

bushel  4  pecks  equals  1  bu. 

second  60  sec.  equals  1  min. 

minute  60  min.  equals  1  hour 

hour  24  hours  equals  1  day 

day  7  days  equals  1  week 

week  52  weeks  equals  1  yr. 

month  12  months  equals  1  yr. 

year  365  days  equals  1  yr. 

rod  16%  ft.  equals  1  rod. 
mile 

ton  2000  Ibs.  equals  1  ton 

GRADE  V 

acre 

section  Work  out  relationships 

board  foot  through  problem  work 

sq.  in. 

sq.  ft. 

sq.  yd. 

sq.   (100  sq.  ft.) 


iiiiiiiiiiiiiiiiiiiiin iiimmmiiimiimmiimmHiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiin 

GRADE  VI 

cu.  in.  Work  out  relationships 

cu.  ft.  through  problem  work 

cu.  yd. 

Cord 

gill  4  gills  equals  1  pt. 

mill  10  mills  equals  1  cent 

barrel 

GRADE  VII 

Degree  of  angle 

degree  of  arc 

minutes  60  sec.  equals  1  min. 

seconds  60  min.  equals  1  degree 

Use  of  tables 


IIIIIIHIIIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 

WHOLE  NUMBER  OPERATIONS 

ADDITION 

In  presenting  addition  one  of  several  plans  may  be  selected  for 
arranging  the  steps  but  in  any  plan  provision  must  be  made  to  teach 
the  oral  and  written  forms,  the  carry,  column  addition  and  special 
points  of  difficulty  such  as  adding  when  O  is  involved,  adding  money 
values,  and  adding  named  objects.  In  all  work  in  addition  the  applica- 
tion to  real  things  should  constantly  be  made.  In  this  connection  the 
basic  law  of  addition  that  only  like  things  may  be  added  should  be 
stressed 

The  only  steps  offering  difficulty  are  those  involving  the  carry  and 
column  addition.  The  carry  idea  can  be  made  clear  to  pupils  if  they 
understand  place  values,  and  are  able  to  make  ah  analysis  of  a  number. 

^ 

STEPS  IN  ADDITION 


Written  Form 


Oral  Form 


Step  I.  The  addition  facts 

Type  Form  Say  or  think 

4  4  and  7  are  11 

+  7 


Suggestions  for 
Rationalization 


Use  objects 


11 


EXERCISES 


5 

+   8 


4 

+  3 


6 
+  8 


4 
+  9 


5 

+  2 


7 
+   3 


1 
+  4 


5 
+   6 


IIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIU 

Step  II.     To  add  a  two-digit  number  and  a  one-digit  number  (with- 
out carry). 


Type  Form 
13 

+  4 


Say  or  Think. 

First  Form 
13  and  4  are  17 

Second  Form 
3  and  4  are  7 
Bring  down  the  1 


Use  objects. 

First  Form  is  to  be  used  in 
most  drill  work.  The  Second 
Form  is  to  be  used  to  show 
method  followed  in  more  com- 
plex exercises. 


12 

+  4 

16 


Extensions. 
First  Type  Form 


22 

+  4 

26 


32 

+  4 

36 


42 
4  etc. 

46 


Say  or  Think.     There  is  no  need  for  ob- 

12  and  4  are  16  jective     rationalization. 

22  and  4  are  26  The     work     is     largely 

32  and  4  are  36  oral  and  the  drill  is  car- 

42  and  4  are  46  ried    on    in    much    the 

etc.  same   way   the    drill   in 

Step  1  was  carried  out. 


Second  Type  Form 

214 
+  342 

556 


Say  or  Think 

2  and  4  are  6 
4  and  1  are  5 

3  and  2  are  5 


Many  exercises  of  this 
type  may  be  given  since 
nothing  new  is  involved 
except  the  new  written 
form.  This  exercise  is 
merely  a  group  of  three 
addition  facts. 


EXERCISES. 


18 

+  1 


2J 


16 
+  3 


12 
+  5 


L6 

•  2 


17 

+  2 


13 

4 


14 

+  3 


13 

+  5 


23 

+  5 


33 
+  5  etc. 


1  1 
3 


24 
+  3 


34 
+  3  etc. 


241 
435 


268 
411 


357 
132 


342 
546 


567 
312 


Step  III.     Column  addition  sums  less  than  10. 
Type  Form 

3 

Say  or  Think 
2  1,  3,  5,  8  Use  objects  if  necessary 

+  !_ 

8 

EXERCISES. 


5 

4 

2 

2 

4 

6 

2 

3 

1 

2 

2 

2 

0 

1 

4 

2 

1 

1 

+  1 

+  1 

+  1 

1 

+- 

H  — 

— 

— 

— 

+  1 

it iiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii itiiiiiiiiiiu 


Step  IV.     The  carry. 


Type  Form 
First 
18 
+     5 


Second 
18 
+  22 

40 

Third 

17 

+  26 

43 


Think 

5  and  8  are  13 
Write  3,  carry  1 
1  and  1  are  2 


Think 

2  and  8  are  10 
write  0,  carry  1 

3  and  1  are  4 


Say  or  Think 
6  and  7  are  13 
write  3,  carry  1 
3  and  1  are  4 


There  is  some  difference  of 
opinion  concerning  the  best 
type  form  to  use  in  opening 
the  work  on  th:  carry  It  is 
thought  to  be  a  good  plan  to 
use  the  first  type  form  for 
many  oral  exercises  of  the 
form 

18         28         38         48  etc. 
5555 

23  33  43  53 
and  follow  this  work  with 
written  exercises  of  the  sec- 
ond type  form.  In  this  form 
the  sum  of  the  units  column 
is  10.  After  these  two  forms 
have  been  used  the  third  form 
may  be  used. 


Extensions 
143 

+       8 

151 

263 
+     14 

277 

435 

+     89 

524 

647 
+  156 

803 

24 
+     16 

40 


Say  or  Think 
11,  5,  1 


Say  or  Think 

7,  7,  2 


etc. 


etc. 


etc. 


It  is  desirable  to  eliminate  all 
unnecessary  language  forms. 
As  soon  as  possible.,  pupils 
should  use  the  forms  suggest- 
ed for  these  exercises. 


14 

8 


16 

+     9 


EXERCISES. 
37 


26 

1 4 


+  23 


+ 


28 
13 


+ 


47 
29 


442 

7 


361 
16 


297 
47 


625 
365 


4*7 
+  603 


276 

-f  434 


iiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiimiiiiiiiN 


Step  V.     Column 
Addition 

Type  Form 
6 
3 
4 

r 

7 

8 

33 


Say  or  think 
15,  20,   27,  33 


Pupils  should  add 
continuously  up  or 
down.  To  allow  pupils 
to  look  for  easy  groups 
develops  a  bad  habit 
and  encourages  pupils 
to  slight  the  more  dif- 
ficult combinations. 


Extensions 

14 

63 

+  14 

91 


Say  or  think 
7,  11  Write  1  carry  1 
2,  8,  9  Write  9 


The  teacher  is  to  use 
her  judgment  as  to  the 
complexity  of  the  exer- 
cises selected. 


126 

14 

+  208 

348 


Say  or  think 
12,  18  Write  8  carry  1 
1,  2,  4  Write  4 
3   Write   3 


EXERCISES 


62 
14 
23 


41 
25 
62 
73 


241 
62 
16 


146 

25 

216 

143 


641 
18 
16 

217 


iiiiiiHiiiiimiiiiiiiiiiiu iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiii i mi iiiHiiiimiii 


Step  VI.     To  add  Money  Values. 


Type  Form 
1st.     $14 

+  $24 

$38 


Say  or  think 

Either 
38,  38  dollars 

or 
38    dollars 


The  new  point  is  the 
use  of  the  sign  $ 


2nd     14c 
8c 

22c 


Say  or  think 

either 
22,  22  cents 

or 
22  cents 


The  new  point  is  the 
use  of  the  sign  c. 
Teacher  to  use  judg- 
ment 


3rd 


$1.14 
2.61 

.08 

$3.83 


Say  or  think 

same  as 

above 


$21 
$62 


$25 
$16 


EXERCISES 


$172 
$25 
$14 


17c 
14c 


25c 
62c 


62c 

14c 

8c 


$1.84 
$  .25 


$4.92 
$1.73 


$  .24 
$4.16 


$6.24 
$  .04 

$4.08 


ADDITION    DRILL   CHART    (20  days) 


6  G  6 
789 

7  7 
8  9 

8 
9 

5  5 

6  7 

5  5 
8  9 

6  6 

7  8 

556 
899 

7  7 
8  9 

8 
9 

5  5 

6  7 

r  6 

7  8 

670 
986 

044 
998 

4  9 

7  9 

8  7 
8  7 

0  3 
8  5 

3  3 

6  7 

316 
886 

033 
794 

2  4 
9  5 

2 
5 

4  4 
9  8 

3  2 
8  9 

2  2 
4  6 

541 
549 

5 
3 

6  8 
4  2 

3  2 
3  2 

222 

673 

2  1 

2  7 

2  3 

7  3 

4  1 
4  6 

876 
999 

6  5 

7  9 

5  0 
8  5 

5  5 

6  7 

6  6 

8  9 

770 
894 

556 

897 

6  7 
9  8 

7  0 
9  3 

8  6 
9  8 

6  5 

7  9 

550 
762 

443 
759 

3  2 
6  5 

t\ 

8 

3  3 
4  5 

3  3 

7  8 

267 
467 

233 
969 

4  7 

7  7 

8 
8 

3  3 
4  5 

A  4 
8  9 

2  2 
4  5 

652 
4  5  Z 

1 

5 

2  3 
2  3 

1 

4 

945 
945 

1 
3 

2  2 
6  3 

£  1 

2  2 

556 

787 

677 
989 

0  2 
1  9 

8  6 
9  8 

5  5 
9  8 

550 
768 

776 
989 

6  5 

7  8 

5 

7 

8  7 
9  8 

6  6 

8  7 

5  5 

9  8 

344 

757 

2  2 

7  8 

6  3 

4  3 

3  3 
4  5 

3  3 
8  9 

6  7 
6  7 

033 
667 

4  4 

8  9 

2 

8 

0  2 
4  9 

3  3 

5  8 

3  2 
9  4 

4  1 
4  1 

8  5 
8  5 

2  1 
C  9 

692 
492 

3  1 

3  8 

2  2 

6  7 

4  5 
4  5 

1 

7 

766 
997 

5  5 

9  8 

5 

7 

8  7 
9  8 

6  5 

8  8 

5  5 
7  6 

766 
997 

5  5 

9  8 

5  0 

7  1 

8  7 
9  8 

6  5 

8  9 

550 
863 

033 
945 

3  4 
7  5 

2  6 
5  6 

0  4 

7  9 

4  4 

8  7 

3  2 
8  4 

233 
945 

3  2 
6  8 

6  7 
4  7 

3  3 

7  8 

3  4 
9  5 

6  8 
6  8 

922 
932 

1 
6 

2  2 

6  7 

3  4 
3  4 

1 
5 

521 
534 

9  2 
9  2 

3  1 
3  3 

776 
989 

6  5 

7  9 

5 

7 

8  6 
9  8 

G  5 
7  8 

550 
762 

766 

898 

6  5 

7  7 

5 
6 

8  7 
9  9 

6  6 

9  7 

550 
989 

044 
557 

4  4 
8  9 

4  4 

8  7 

3  3 

9  7 

2  2 

4  8 

023 
794 

3  3 
5  6 

2 
6 

3  3 

7  8 

3  4 
9  5 

2  2 
6  7 

222 
567 

4  5 
4  5 

6  2 
4  3 

2  1 

o  -j 

2*  7  3 
773 

4  1 
4  9 

8  5 
8  5 

2  1 
3  8 

1 
2 

iiimiiiim iiiiiiiiiiiimiim mi i iiiiiiiiiiiiiiiiiiiniiiiiiiiii mini i i i iiiiiiiiiiiiiiiiiiiiiiiimntiiiiiiiiiiiiiiiiiii 

SUBTRACTION  OPERATION 

All  number  processes  are  based  upon  a  few  simple  principles.  The 
following  outline  shows  some  of  the  simple  principles  upon  which  the 
four  operations  in  whole  numbers  are  based: 

1.  Quantity 

1.    A  quantity  may  be 

A.  Increased 
(by  addition) 

(by  multiplication) 

B.  Decreased 

(by  subtraction) 

C.  Divided 

(by  division) 

2.  Two  or  more  like  quantities  may  be 

A.  Combined 
(by  addition) 

B.  Compared 

(by  finding  the  difference— subtraction) 
(by  measurement — division) 

It  is  evident  that  the  process  of  subtraction  is  based  upon  the  idea 
of  decreasing  a  quantity  and  upon  the  comparison  of  two  quantities  by 
finding  the  difference. 

We  express  these  ideas  as 

The  take-away  idea  (decreasing  a  quantity) 
The  difference  idea  (comparing  two  quantities) 

We  also  have 

The  adding-to  idea  (as  used  in  making  change) 
This  last  idea  is  not  based  on  a  fundamental  number  principle  as 
outlined  above.    It  is  an  outgrowth  of  the  connection  between  the  addi- 
tion and  subtraction  facts. 

Three   Illustrative  Examples — 

Elsie  has  5  apples  and  loses  2  apples.  How  many  has  she  left? 
(The  thought  is  2  from  5.) 

Elsie  has  5  apples  and  John  has  2  apples.  Who  has  more  apples? 
(The  thought  is  comparison  by  difference.) 

Elsie  has  2  apples.  How  many  more  must  she  obtain  to  have  5? 
(The  thought  is  what  must  be  added  to  2  to  make  5.) 

Since  any  one  of  these  three  thoughts  may  come  to  the  pupil  it  is 
not  desirable  that  the  teacher  should  use  one  only  and  submerge  the 
other  two. 

To  further  illustrate: 

5  Pupil  may  say  or  think 

2  2  from  5=3  (thinking — take  away) 

5  Pupil  may  say  or  think 

2  The  difference  between  5  and  2=3         (thinking— difference) 

5  Pupil  may  say  or  think 

2  2  and  3  are  5  (thinking — adding-to) 


iiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiimiiiimiiiiiiiiiiiiiiimiiimmiiiiM 

The  statement  of  the  problem  always  suggests  the  thought  form 
and  in  turn  the  thought  form  always  suggests  the  language  form. 

Certain  complications  always  arise.  For  example,  a  pupil  may  think 
"What  is  the  difference  between  5  and  2?"  and  use  the  adding-to  method 
to  discover  the  difference.  Or  a  pupil  may  think  "2  from  5  are  how 
many?"  and  use  the  adding-to  method  to  discover  the  answer.  Or  still 
again  the  pupil  may  think,  "2  and  how  many  are  5?"  and  use  the  take- 
away method  to  discover  the  result. 

Since  the  learning  of  the  addition  facts  automatically  gives  the 
pupils  the  subtraction  facts,  it  is  desirable  that  pupils  use  this  informa- 
tion. It  is  desirable  to  make  every  possible  use  of  the  fact  knowledge 
acquired  through  a  study  of  the  addition  facts.  It  is  not  desirable, 
however,  to  submerge  entirely  the  take-away  thought  and  the  difference, 
thought  simply  because  the  adding-to  method  provides  a  possible  way 
to  discover  a  result.  In  all  problem  work  the  conclusion  should  always 
be  stated  in  terms  of  the  question  that  is  asked. 

As  the  difficultly  of  examples  and  problems  increases  it  becomes 
necessary  to  develop  a  mechanical  process  to  aid  the  mind.  Two  me- 
chanical processes  are  in  common  use.  The  following  examples  illus- 
trate them: 

81  9  and  2  are  11  81  9  and  2  are  11 

39  4  and  4  are  8  39  3  and  4  are  7 

42  42 

In  the  first  example  3  in  the  subtrahend  is  increased  by  one.  In 
the  second  example  8  in  the  minued  is  decreased  by  one. 

Both  processes  may  be  explained  if  the  teacher  feels  that  it  is 
necessary  to  make  explanation. 
81    =    8  tens  1  unit        =        8  tens  11  units 
39    =     3  tens  9  units      =        4  tens  9  units 


81 
39 


8  tens  1  unit 
3  tens  9  units 


4  tens  2  units 

7  tens  11  units 
3  tens  9  units 


10  units  have  been  add- 
ed to  the  minuend  and 
1  ten  has  been  added  to 
the  subtrahend. 


The  minuend  has  been 
written  in  an  equivalent 
form. 


Results  from  careful  testing  fail  to  show  that  one  mechanical  pro- 
cess produces  better  results  than  the  other.  It  seems  to  be  the  general 
feeling  among  those  who  have  given  the  question  attention  that  either 
mechanical  process  may  be  adopted. 

It  is  the  feeling  of  teachers,  however,  that  only  one  mechanical  pro- 
cess should  be  taught. 

This  does  not  mean  that  only  one  thought  and  language  form  must 
be  used  for  all  problems. 

The  following  outline  shows  some  possible  language  forms  with 
each  mechanical  process. 


81 
39 

42 

81 
39 

42 

81 
39 

42 


Say  or  think 
9  and  2  are  11 
4  and  4  are  8 


9  from  11  are  2 
4  from  8  are  4 


The  difference  between 
11  and  9  is  2 
The  difference  between 
8  and  4  is  4 


81 
39 

42 

81 
39 

42 

81 
39 

42 


Say   or   think 
9  and  2  are  11 
3  and  4  are  7 

9  from  11  are  2 
3  from  7»  are  4 


The    difference 
11  and  9  is  2 
7  and  3  is  4 


between 


miimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin^ 

The  general  feeling  is  that 

1.  Teacher  should  select  one  mechanical  process. 

2.  Teacher  should  use  but  one  language  form  during  that  first 
stage  of  instruction  when  children  are  developing  a  skill  in  the  per- 
forming of  the  mechanical  process. 

3.  Teachers  should  use  all  three  language  forms  after  the  skill 
is  acquired 

4.  Teachers  should  always  allow  pupils  to  express  the  conclusion 
in  a  problem  in  terms  of  the  question. 

It  would  seem  that  teachers  and  superintendents  should  take  a 
reasonable  attitude  toward  the  question  of  teaching  subtraction.  Too 
much  time  is  wasted  in  argument  that  has  no  base  except  the  per- 
sonal prejudice  of  this  or  that  person. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiH 

The  following  outline  of  steps  shows  a  possible  choice  of  mechani- 
cal process  and  language  form: 

Step.  I  Oral  Form  Suggestions  for  Bxplan- 

Written  Form  ation. 

65  48  27  Build  on  the  addition 

-3  -5  -4  and .  subtraction    facts. 

The  purpose  in  this  drill 

62  43  23  is  to  develop  a  written 

form  and  the  use  of  the 
subtraction  facts  in  con- 
nection with  the  larger 
numbers. 


73 
—  2 


EXERCISES 

95 
i 


38 

—  7 


97 

—  5 


39 
-  2 


54 


54 
—  4 


47 
—  1 


69 

_          O 


Illllllllllllllllllllllllllllllll! llllllllllllllllllllllMlllllllllllllllllllllllllllllllimilllllHIIIHtllllllllllllllllllllllllllllllllllllllllllllllN 

2— To  take  a  one  digit  number  from  a  two  digit  number: 
Written  Form.  Say  or  Think.  Suggestions. 

23        42        74        20  Same  as  above.    Build 

_5        _9        -4  on  addition  and  subtrac- 

tion  facts  and  upon  the 

16  65        is  series  drill  in  addition. 

Perfect  the  written 
forms.  Much  time 
should  be  given  to  drills 
of  this  type. 

EXERCISES 

83  42  55  22  35  46 

_  5  _3  _6  —  4  -6  -7 


21  34  43  61  72 

—  6  —  5  —  2  -4 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin 

3 — To  take  a  two  digit  number  from  a  two  digit  number: 

Written  Form  Say  or  Think.  Suggestions. 

34  9  and  5  are  14,  Allow   pupils   to   per- 

— 19  write  5,  carry  1.        feet  the  mechanical  op- 

—  2  and  1  are  3,          eration.      Any   explana- 

15  write  1.  tion   that   is   attempted 

should  be  given  with 
the  thought  of  simply 
making  the  operation  a 
reasonable  one. 

EXERCISES. 

88  75  93  86  27 

—19  —16  —14  —27  —16 


55  84  72  52  49 

—29  —18  —68  —19  —26 


iiiiiiiiiiiiiiini iiiiiiiiiiiiimiimiiii iHiiiiiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii imiiiiiimiimiiiiimHiiiiiiiiiiiiiHiMiiiiiiiini 


4— An  extension  of  Step  3. 

Written  Form.  Say  or  Think 

416  9  and  7  are  16, 

write  7,  carry  1. 
8  and  3  are  11, 

237  write  3,  carry  1, 

2  and  2  are  4, 
write  2. 


325 
-186 


EXERCISES. 

387 
—198 


Suggestions. 
Perfect  the  mechani- 
cal operation.  Make  the 
operation  as-  reasonable 
as  possible  without  at- 
tempting rigid  proofs. 
Build  on  the  addition 
and  subtraction  facts. 


783 
—595 


476 
-297 


323 
-198 


456 
-179 


603 
-494 


805 
167 


234 
156 


iiiiiiiiimiiiHinuiiiiiiiiiiiHiiifiiiiHiiiiiiimimiiiiiiiimiiiim 

5 — To  take  any  number  from  any  other  number: 

Written  Form.  Say  or  Think.  Suggestions. 

4123 
—1576  Same  as  above.  Same  as  above. 

EXERCISES. 

3781          6328          5342 
—1889        —4789        —3457 


2776          7543          3586 
-1889        —2654        —1798 


5786          4080          7006 
3877        —2399        —3687 


Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllim 

6— MONEY  VALUES 

14c  $32  $2.15 

7c  $16  $1.67 

7c  $16  $  .48 

EXERCISES 

13c        27c        86c        64c 
8c        13c        32c        19c 


$42        $19        $90        $88 
$18        $  8        $  6        $32 


$1.76  $9.49  $10.92 

$  38  $3.46  $  6.48 


iiiiniiMmiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim 

MULTIPLICATION  OPERATION. 

1.  Definition  of  multiplication  as  an  addition  process 

2.  Definition  of  X  sign  (to  be  read  times). 

3.  Definition  of  terms  Multiplicand,  Multiplier  and  Product. 

4.  Fundamental  law  that  the  multiplier  must  be  abstract  and  that 
the  product  is  of  the  same  kind  as  the  multiplicand. 

STEPS. 

1.     To  multiply  a  two-digit  number  by  a  one-digit  number  (without 
a  carry). 

23  Say  or  think: — 2  3's  are  6,  write  6 

x2  2  2's  are  4,  write  4 

46 

This  step  may  be  extended  to  the  multiplication  of  a  number  of  3, 
4  or  more  digits  by  a  one-digit  number  (without  a  carry) . 
223  14243 

x  3  x       2 

669  28486 

Argument — Base  the  operation  on  the  multiplication  facts,     The 
purpose  of  this  step  is  largely  that  of  perfecting  the  written  form. 

EXERCISES. 


2  x 

24 

33 

443 

x3 

x2 

3  x 

31 

21 

321 

4  x 

12 

x3 

x2 

2  x 

43 

44 

2134 

x2 

x2 

3  x 

23 

32 

1243 

x3 

x2 

4  x  21 


iiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiiiiiN 

-.     To   multiply  a  two-digit  number  by  a  one-digit  number    (with 
a  carry). 

43         Say  or  think 
x  4         4 — 3's  are  12,  Write  2.  carry  1 

4 — 4's  are  16,  16  plus   1   are  17,  write   17 
172 

To   multiply  a  three-digit  number  by  a  one-digit   number   (with  a 

carry » . 

456         Say  or  think 

x  f>         5 — 6's  are  30,  Write  0,   carry  3 

5— 5's  are  25,  25  plus  3  are  28,  write  8,  carry  2 
2280         5— 4's  are  20,  20  plus  2  are  22,  write  22 

This  step  may  be  extended  to  the  multiplication  of  a  4,  5.  or  more 
digit  number  by  a  one-digit  number  (with  a  carry). 

6434  56243 

x     7  x       6 


45038  337458 

Argument — The  base  for  the  explanation  is  number  analysis  and 
knowledge  of  place  value.    Objects  may  be  used  if  necessary. 
4::         4  tens  3  units 

4   =  4 


16     "  12  =17  tens  2  units  =  172 

EXERCISES 


55 

x2 

343 
x4 

456 
x3 

3251 
x4 

16024 
x4 

67 

\:: 

6203 

x5 

41292 
x7 

37 

x4 

648 
x2 

6213 

x5 

23125 
x8 

iiiiiiiiiiiiiiiiiiiiiiiiiiimiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiwiiiimiiiiiiiiiim 

3.     To  multiply  a  two-digit  number  by  a  two-digit  number. 

24  Say  or  think: — 6 — 4's  are  24,  write  4,  carry  2 

x!6  6— 2's  are  12,  plus  2  are  14,  write  14 

144  1 — 4  is  4,  write  4  under  tens 

24  1—2  is  2,  write  2,  add 

384 

This  step  may  be  extended  to  the  multiplication  of  3,  4,  or  more 
digit  numbers  by  a  two-digit  number. 

As  in  addition  and  subtraction  all  work  should  be  connected  with 
real  problems  as  soon  as  the  mechanical  skill  is  acquired. 

Argument — The  new  point  in  this  step  is  the  set-over.    The  teacher 

O  A 

may  show  that  .,*  means  6  x  24  and  10x24 
lo 

6  x  24  =  144 
10  x  24  =  240 

Sum  of  products— 384 

This  step  may  be  extended  as  follows: 

462  8434 

x  14  x      49 

1848  75906 

462  33736 

6468  413266 

EXERCISES. 

24          345          3845 
x!7          x!7          x  36 


37          426          6271 
x!2          x!3          x  45 


63          509          3678 
x!5          x23          x79 


inn iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiii iiiiiiiiiiiiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiimiiiiiiiiiiiiiiiiiiiiiiiiiii 

4.— To  multiply  a  number  by  a  three  digit  number: 

432        Say  or  think: 
x  178         8 — 2's  are  16,  write  6,  carry  1 

8— 3's  are  24,  plus  1  are  25,  write  5  carry  2 
:',456         8— 4's  are  32,  plus  2  are  34,  write  34,  etc. 
3024 
432 


76896 

This  step  may  be  extended  to  multipliers  of  4,  5  or  more  digits  as 
semis  advisable. 

EXERCISES 


365 

x!27 

3264 
x!568 

35792 
X14653 

28765 
X10531 

39607 
X52613 

432 
x!29 

6538 
x!296 

467 
x235 

3729 
x!076 

178 
x!26 

4291 
x2305 

429654 
X132578 

237 
x964 

3529 
x!264 

7960374 
X1207123 

IIIIMimilllllllllllllllllllllllllllllllllllllllllllllllllllllMIIIIIIIIIIIIIII Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllliii 

5 — To  multiply  money  values: 

$46.34 
25 


23170 
9268 

$1158.50 

The  language  form  would  be  the  same  as  in  previous  steps.  The 
point  in  the  product  should  be  relatively  in  the  same  position  as  it  is  in 
the  multiplicand. 


$25.13 
x!3 


$82.96 

x27 


EXERCISES 

$365.27 
x25 


$387.91 
x79 


$4216.17 
x36 


$3781.27 
x25 


$35.68 
x35 


$525.75 
x67 


$4291.21 
x206 


$38.92 
x!25 


$57.68 
x236 


$468.18 
x213 


$721.46 
xl-75 


$7691.25 
x357 


$8219.14 
x  1210 


MULTIPLICATION    DRILL   CHART    (20   days) 


81    32    16 

54    24    12 

72    28    16 

49    24    12 

72    28    15 

49    21     9 

64    27     6 

48    21    15 

64   100    10 

48    70     8 

63    45     4 

42    30    10 

63    90 

42    60 

56    40 

81    25 

56    80 

81    50 

54    35 

72    20 

36    18 

27    14 

36    18 

32    14 

64    24    16 

48    28    16 

23    24    16 

42    28    12 

63    21     9 

42    27     6 

56    21     4 

81    27     6 

( 

56    45     8 

81    50     4 

54    60    8 

12    40     9 

54    35 

72    70 

49    80 

64    30 

49    25 

64    90 

48   100 

63    20 

32    18 

36    18 

32    18 

32    14 

56    28    16 

81    27    12 

54    28    16 

72    30    12 

54    36    10 

72    32    15 

49    27    10 

63    64     6 

49   100    15 

64    70     9 

48    45     8 

64    56     4 

48    90 

63    60 

42    40 

24    25 

42    80 

56    50 

81    35 

36    20 

21    18 

24    14 

32    18 

21    14 

49    32     6 

64    27    12 

48    27    16 

63    28    12 

48    50    10 

63    28     4 

42    21     8 

56    24     4 

42    70    36 

56    60     9 

81    45    15 

54    30    16 

81    90 

54    80 

72    40 

49    25 

n   is 

49   100 

64    35 

48    20 

21    16 

24    14 

32    18 

36    14 

42    32    30 

56    36    45 

81    27    90 

54    36    80 

81    27    20 

54    28    50 

72    24   100 

49    32   100 

72    21    18 

49    24    16 

64    21    14 

48    28    12 

64    50     8 

48    25    10 

63    70     4 

42    45     8 

63    40     9 

42    35    15 

56    80     9 

81    50    15 

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiin 


DIVISION  OPERATION 

1.     Definition  of  division  as  partition  and  also  as  measurement. 
2  apples 


2  |  4  apples 
2 


Partition. 


Measurement. 


5c  |  lOc 

In  partition  the  thought  is  to  divide  a  quantity  into  parts. 
In  measurement  the  thought  is  to  measure  a  quantity  by  another 
quantity  of  the  same  kind. 

2.  Definition  of  the  -j-  sign  (to  be  read  divided  by). 

3.  Definition  of  the  terms  dividend,  divisor,  quotient. 


Partition. 

2  oranges 
3  |  6  oranges 

5  yards 


2 


10  yards 

4  trees 
3  fT2  trees 

2  marbles 


4  |  8  marbles 

2  dollars 
2  fTdollars" 

2  miles 


Measurement. 
5 


2  yds.  |  10  yards 

_2 

3  books  |  6  books 

2 


2  chairs  |  4  chairs 

_2 

2c  " 


4c 

4 


2  pencils      8  pencils 


5  |  10  miles 


Illllllllllllllllllllllllllllllllilllllllllllllllllllll lllllllllllllllllllllllllllltlltlllMIMIIIIIIMIMIIIIIIIIIIIIIIIIllllllinillllllMIIMMIIIMIIIIMIIIIIIillMMIIIIIIIIIIIIIIIII 

STEP  I. 

1.  To  divide  a  two-digit  number  by  a  one-digit  number  (without  a 
carry  and  without  a  remainder). 

23         Say  or  think:   Ihere  are  2— 2's  in  4,  write  2 
2  |~~46 There  are  3 — 2's  in  6,  write  3 

This  step  may  be  extended  to  cover  the  division  of  3,  4  and  more 
digit  numbers  by  a  one-digit  number  (without  a  carry  and  without  a 
remainder). 

123  2134 

3  |  369  2  |  4268 

The  short  division  work  is  to  be  extended  as  follows: 

2  62  72 

2  ]  124  2  |  144 

3.  Say  or  think 

37  Ihere  are  3 — 2's  in  7  and  1  over 


2  j  74  There  are  7— 2's  in  14 

4.  Say  or  think 

32  There  are  3 — 2's  in  6 

2  |  65  Rem.  1     There  are  2 — 2's  in  5  and  1  over 

Note:— Later  the  form         32M>  may  be  taught. 

2  |~65~~ 

5.  Say  or  think 

37  There  are  3— 2's  in  7  and  1  over 


2  |  75  Rem.  1     There  are  7— 2's  in  15  and  1  over 

There  aro  4     4'e  in  17,  tin<M-mror,  WFito- 

•4-eftrryl. 
There  are  4 — 4ra  ill  10,  wiite4r 

Thfs  step  may  be  extended  to  cover  the  division  of  3,  4  and  more 
digit  numbers  by  a  one  digit  number  (with  a  carry  and  with  a  re- 
mainder). 

14410  —  3  over 


57643        Say  or  think: — There  is  one  4  in  5  and  1  over,  write  1, 


3  |  93  2  |  468  2  |  148 


4  |  84          3  |  3963          3  |  129 


imiiimiiiiiiiiiiiiiiiimiimiiiiiiiiiiimiimiiiiimiiiiiiiiiiiiiiiiiiiiiM 

STEP  II. 

The  written  form  in  long  division  is  to  be  mastered  through  the 
use  of  divisors  11  and  12.  Pupils  have  learned  the  product  facts  so 
the  attention  may  be  given  to  the  written  form. 

11 


11  j  121         Say  or  think — -12  contains  11  once,  write  11. 

11  subtract,  bring  down  1,  11  contains 

11  once. 
11 
11 


12 


11  j  132         Say  or  think — 13  contains  11  once,  write  11, 

11  subtract,  bring  down  2,  22  contains 

11,  2  times. 
22 
22 


12 


12  |  144         Say  or  think— 14  contains  12  once,  write  12, 

12  subtract,  bring  down  4,  24  contains 

12,  2  times. 
24 
24 


EXERCISES. 


11  |  121  12  j  108 

11  f  132~  12  ("132" 

11  ["110"  12  j~14T 


Illlllllllllllllllllllllllllllllllllllllltllllllllllllllllllllllllltllllllllllllllllllllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 

STEP  III. 

The  second  step  may  be  extended  to  include  dividends  not  known. 
(No  remainder.) 

22 


12  |  264         Say  or  think— 26  contains  12,  2  times,  write  24, 


24 


24 
24 


subtract,  bring  down  4,  24  contains 
12,  2  times. 


21 


Say  or  think— 23  contains  11,  2  times,  write  22, 


22 


subtract,  bring  down  1,  11  contains 
11  once. 


Etc. 
EXERCISES. 


11      231 


12  I  264 


11      164 


11      143 


12  |  286 

12  lies' 


11      187 


12  I  516 


11  |  297 


11      594 


11  |  396 


12  |  408 
12  ]~504~~ 
12  f~372~ 


iiiiiiiiiiiiMiimiiiiiiiiiHiiiiiiiiiiiiiiiiimimmiiiiiiiiimiiiiimiimm 

STEP  IV. 

The  next  step  should  be  that  of  using  2  digit  divisors  not  used  in 
connection  with  the  multiplication  facts. 

11 


29  |  319        Say  or  think— 31  contains  29,  once,  write  29, 

29  subtract,  bring  down  9,  29  contains 

29,  once. 
29 
29 

13 


169        Say  or  think — 16  contains  13,  once,  write  13, 
13  subtract,  bring  down  9,  39  contains 

13,  3  times. 

39 

39 

21 


14  |  294        Say  or  think — 29  contains  14,  2  times,  write  28, 
28  subtract,  bring  down  4,  14  contains 

14  once. 
14 
14 


Note — The  divisors  should  be  selected  carefully;  21,  22,  31,  32,  etc., 
should  be  used  first  since  the  quotient  digits  are  more  easily  deter- 
mined. 

EXERCISES. 


15  |  315 


17      355 


24      312 


13  j  273 
35  ("630 


27 
24 

ni 

14 
18 

|  837 
]~26F 
|  641 
|  448 

28  |  364 
33  p693~ 

42  |  482 
16  |  336 

|  270 

23  |  483 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIN 

STEP  V.— LONG  DIVISION. 


203 

34  I  6902 
68 

102 
102 


Say  or  think— 69  contains  34,  2  times,  write  68  (2x34), 
subtract,  bring  down  0, 10  does  not  con- 
tain 34,  write  0,  bring  down  2,  102  con- 
tains 34,  3  times. 


Long  division  should  be  developed  slowly  with  care  given  to  proper 
language  forms  and  to  the  order  of  steps  in  the  mechanical  operation. 


226 


28 


6342 
56 

74 
56 

182 
168 


Say  or  think — 63  contains  28,  2  times,  write  56  (2x28), 
subtract,  bring  down  4,  74  contains  28, 
2  times,  write  56  (2x28),  subtract,  bring 
down  2,  182  contains  28,  6  times,  write 
168  (6x28),  subtract,  quotient  226,  re- 
mainder 14. 


14  remainder. 
The  extension  to  the  3  digit  divisor  may  be  made  without  difficulty. 

EXERCISES. 


35  |  7105 
24  |  7488 


27  |  8127 


31  I  6386 


33      6421 


49  |  4982 
63  |  1348 


72  |  7505 


29  |  4209 


61  |  8298 


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STEP  VI.— MONEY  VALUES. 
$.21 


7  j  $1.4jf        1.     Place  point  over  the  point  in  the  dividend. 

$3.21 
3  |  $9.63        2.     Proceed  in  the  usual  way. 

$1.14 

13  I  $14.82 
13 


18 
13 

52 
52 


Note — Do  not  attempt  to  teach  decimal  fractions  at  this  time.  Re- 
fer to  these  numbers  as  expression  of  money  values.  Limit  the  work 
to  simple  cases. 

EXERCISES. 

.81       12  |  $14.40 
$1.55       13  r$3lT69 


6  |  $1.26 


11  I  $13.20 


14  |  $29.40 


$6.09 


15  |  $75.45 


17  I  $39.61 


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FRACTIONS 

INTRODUCTION 

1.  To  recall  that  a  quantity    may    be    divided    into  >  equal    parts 
(use  objects). 

2.  To  define  a  unit.  (Use  objects) 

3.  To  define  a  fraction  as  one  or  more  of  the  equal  parts  of  a  unit 
}-  means  4  of  the  5  equal  parts  of  one  thing 

(%  of  a  pie.) 

4.  To  show  the  distinction  between  a  unit  and  a  number. 

5.  To  define  a  fraction  as  one  of  the  equal  parts  of  a  number  $.-, 
means  1  of  the  5  equal  parts  of  4  things.  (%  of  4  pies). 

6.  To  show  that  %  of  1=4.S  of  4  (Use  drawings  or  objects.) 

7.  To  show  that  the  left  over  part  when  a  measurement  is  made 
is  the  fractional  part  of  the  unit. 

8.  To  show  that  the  fraction  sometimes  is  used  to  represent  a 
division.    %  means  4-^5. 

9.  To  show  that  this  last  meaning  is  connected  with  the  defini- 
tion suggested  in  No.  5. 

10.     To  define  words  numerator  and  denominator  and  to  perfect  all 
written  forms. 

11.  To  show  that  under  the  definition  given  in  No.  3  the  denom- 
inator shows  the  parts  into  which  the  unit  is  divided  and  the  numer- 
ator shows  the  number  of  parts  used. 


nMHiiimmiiiiiiiiiiiiiiiiiiHiiiiiiiiimiiiiiimiiiimiiiiiiiimiiiimiiiim 

EQUIVALENT  FORMS. 

1.     To  show  that  a  whole  number  may  be  expressed  as  a  fraction. 

(a)  1=%=%       etc.     Use  objects  or  drawings. 

(b)  3=4=1%      etc. 

5^i(j£=i^    etc.     Use  objects  or  drawings. 

EXERCISES 

4,  8,  6,  9, 
12,                     7,                   11,                   10, 

5,  14.  16,  18, 


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2.     To  show  that  a  fraction  may  be  changed  to  an  equivalent  form 
by  multiplying  both  numerator  and  denominator  by  the  same  number. 

(a)  }£=-%--...%»  etc.    Use  objects  or  drawings. 

Argument—  If  the  unit  is  divided  into  twice  as  many  parts,  each 
part  is  half  as  large.    Therefore  twice  as  many  parts  are  used. 

(b)  %=d%=%2,  etc.    Use  objects  or  drawings. 
Argument  —  same  as  above. 

(c)  5/s=1%=1%,  etc.    Same  as  above. 

(d)  Establish  rule. 
(c)     Drill. 

EXERCISES. 


% 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM 

3.     To  show  that  a  fraction  may  be  changed  to  an  equivalent  form 
by  dividing  both  numerator  and  denominator  by  the  same  number. 

(a)  %—  }£     Use  drawings  or  objects. 

(b)  %=%     Use  drawings  or  objects. 

(c)  %=%     Use  drawings  or  objects. 


Argument  —  If  the  unit  is  divided  into  half  as  many   parts   each 
part  is  twice  as  large.     Therefore  only  half  as  many  parts  are  used. 

C.  —  Establish  rule. 
D.—  Drill. 

Note:  —  This  is  the  base  for  cancellation.     Since  cancellation  is 
used  later,  the  teacher  should  present  this  process  very  carefully. 

EXERCISES 
(a)     %=%  (b)     %=%  (c) 


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4  To  change  several  fractions  to  equivalent  forms  having  like 
denominators. 

(3)  l._,          1;;          1,      tO     <•',,         %>         %0 

I'se  rules  established  above.    Verify  if  necessary  with  objects. 

Note:  —  This  step  requires  a  discussion  of  the  meaning  of  a  multi- 
ple and  practice  in  the  selection  of  the  common  denominator.  This  is 
the  time  to  develop  the  idea  of  the  lowest  common  multiple.  Only 
enough  time  should  be  devoted  to  the  topic  to  make  possible  the  above 
operation. 

(b)       %      %      %   tO   l%o      15^      Sfa 

Use  rules  established  above.     Verify  if  necessary  with  objects. 

(C)       %      %      %    tO    1<>:,()       l%o       7%(, 

same  as  above 

Note  —  Since  the  need  for  this  process  is  in  the  preparation  of  frac- 
tions for  addition  and  subtraction  it  is  desirable  to  present  this  work 
when  the  pupils  feel  a  need  for  it. 

(d)     Develop  procedure. 

EXERCISES 
(a)     %,    %,    %, 


',- 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 

5.     To  change  an  improper  fraction  to  a  mixed  number. 

(a)  -%=--lV4         Use  objects  or  drawings.  >v\ 

Argument — There  are  %  in  1.     Therefore  there  is  one  unit  in  % 
with  14  over. 

(b)  %=2%        Use  objects  or  drawings. 

Argument — In  1  there  are  %.     Therefore  there  are  2  units  in  % 
with  %  over. 

(c)  Develop  rule — Divide  numerator  by  denominator. 

EXERCISES 

(a)  3/2=ii/2  (b)  %=2i/4 

%--?  %=?  !%='?  %=?  %=? 

%=?  9fr=?  %=?  %-?  %=? 


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6.     To  change  a  mixed  number  to  an  improper  fraction. 
<3"SL;-(iL)^Al%—('%  and  }£)=%.    Use  objects  or  drawings. 

Argument. — In  cne  there  are  %.      ?/•>  and  \(>  are  %. 
(b)     3%=  (%  and  %)=H$. 

Argument— In  1  there  are  %.    In  3  there  are  3  X  %=%• 


(c)  Develop  rule — Multiply  the  whole  number  by  the  denominator. 
Add  the  numerator  to  this  product.  Write  the  sum  over  the  denom- 
inator. 

EXERCISES. 


I'-, 

Us 


31 , 


1%    : 


5%= 
6%= 

99.';= 


Illlllllllllllllllllllllllllllllllllllll Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 

7.     To  change  several  mixed  numbers  and  fractions  to  equivalent 
fraction«;having  like  denominators. 

y2,  %,  3%  Use  objects  or  drawings  if 

<yv>i  i?yl2>  4<fe  necessary. 

EXERCISES 

V3;   %;    %  y12;    9/12;    3o/12 

an;   %;    2%  1%;    2%;   % 

%;   %;   17/io  %o;   4%;   % 

1%;     3%;     2%  1%;     %;     3^ 

2%;    %;   Me  7/i2;    1%;    2% 

%;    %;    i6/i4  %;    1%;    2% 


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ADDITION. 
1.     To  add  unit  fractions  having  like  denominators.  * 

4=1  ],£+!,£  +  }£=%:=  i  Use  drawings  or  objects 


Argument.  —Only  like  things  may  be  added.  Since  the  denomina- 
tor of  a  fraction  shows  the  kind  of  unit,  it  follows  that  fractions  are 
like  when  the  denominators  are  like. 

1  half 
1  half 

l'  halves. 

EXERCISES 

+  .  %     +     K  Vl     +     Vi     +     Vl     +    Vi 


+       14       +       ^ 
+      V,      +      V:>. 


iiiiiiiiiimiiiiiiimiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiimim 

2.     To  add  fractions  having  like  denominators. 

(a)  %  +  1/0  =  %  Use  drawings  or  objects. 
Argument  —  Same  as  above. 

2  fifths 
1  fifth 

3  fifths 

(b)  %  +  %  =  94  T  2 
Argument  —  Same  as  above. 

Note  —  It  is  desirable  to  use  the  following  written  form  as  well  as 
the  forms  shown  above. 


=  2 

EXERCISES. 


3/T      +      %      =  +      Vl  +      % 


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3.     To  add  mixed  numbers  having  fractions  with  like  denominators 
(no  carry). 

2  1,4 

+     3J4 

Remarks — This  may  be  made  clear  by  arranging  the  parts  as  fol- 
lows: 

2  and  14 

3  and  14 

5  and  %  5%    =    &£ 

EXERCISES. 
3%  1^  4^  7% 


7V* 


miiiiiiimmiiiiiiiiiiiiimmiiiiiiiiiimiiiiiiiiimiimiiimiimmniiiiimiim 

4.     To  add  mixed  numbers  having  fractions  with  like  denominators 
(with  carry). 

2% 

+     4ft 

«%    -     7% 
Remarks — }5    =       Ifo.    Add  6  and  1. 

EXERCISES. 

3%  52/r>  6% 

4%  -4%  77/8 


91/2  2i/4 

3%  53/4 


5%  4% 

6%  2% 


Illllllllllimilllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllim 

5.     To  add  unit  fractions  having  unlike  denominators. 
&    =    .% 


Argument — Only  like  things  may  be  added.     Change  fractions  to 
equivalent  forms  having  like  denominators,  then  add. 

EXERCISES. 
V*  Vi  % 


+     %  +     K  +     % 

%  %  % 

+     %  +     %  +     % 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiim 

6.     To  add  fractions  having  unlike  denominators. 


17/12      = 

Argument  —  Same  as  above. 

EXERCISES. 

%  % 


V*  % 

+    %        '  +    %          + 


immniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiimiiiimniiH 

7.     To  add  mixed  numbers  having  fractions  with  unlike  denomina-. 
tors  (no  carry). 

3%    = 


EXERCISES. 

4%  41/6  2y4 

+     31/6  +2%  +     1% 


3^  31/g 

+     5%  +     2% 


+     7%  +     3y4  +     3% 


iiimiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiimiHiiiiMim 

8.     To  add  mixed  numbers  having  fractions  with  unlike  denomina- 
tors. 

4%     —     49^ 
+     2%     =     21945 


EXERCISES. 

2%  5%  47/6 

+     8%  +-   78^0  +     9% 


7%  35^ 

+     »%  +     4%  -f 


llllltllillllllllllllllllllllllllllllllllllllllllllllllllllllllllllHIIIIIIIMIIIIIIIIIIlllllllllllllHIIIIIIIIHIIII IIIIHIIIIIIIHIHIIIIIIIIIIIIIIIIIinillll IIIIIIMIIIIIMIMIIM 

SUBTRACTION. 

1.     To  subtract  fractions  having  like  denominators. 
Use  objects  or  drawings. 
Language  Forms. 

%  1.    %  and  %  are  % 

^  2.     y,  from  %  leaves  % 

3.    The  Difference  between  %  and  %  ls% 

Argument — Similar  to  that  used  in  addition. 
EXERCISES. 


%      = 
—    %      — 


Mo          HB    — 


-    7/i2    = 


Mo 
1H 

1 


-     Ve 


iimimiHiHiimiiiiiiwiiiwiiiiiiiitHitiimiiimiiwiMiiiiiiwiiimiiimiiiiiHim 

2.     To  subtract  mixed  numbers  having  fractions  with  like  denomi- 
nators. 

Language  Forms. 

4%  1.  2->  and  %  are  $.v  2  and  2  are  4- 

2%  2.  $-,  from  %  leave  $'•„  2  from  4  leaves  2. 

3.  The  difference  between  %  and  %  is  %.    The 

2%  difference  between  4  and  2  is  2. 

Argument  —  Similar  to  that  used  in  addition. 
Use  objects  and  drawings. 

EXERCISES. 

2^  7%  8% 


3%  5.}:, 

1  -     4% 


ft  6%  4% 

5%  4-%  37^0 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiihiiii 

3.     To  subtract  fractions  having  unilke  denominators. 


Argument — Similar  to  that  used  in  addition 
EXERCISES. 

%g/  ___      gy 

/9  70 


5/7 


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4.     To  subtract  mixed  numbers  having  fractions  with  unlike  de- 
nominators (no  borrow). 

4%    =    4i%o 


Argument— Similar  to  that  used  in  addition. 

4%  6%  7% 

-2%  -2%  -  5% 

3%  47/8  4% 

1%  2%  -  3% 

4%  5%  5% 


MiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiM 


5.     To  subtract  mixed  numbers  having  unlike  denominators  (with 
borrow). 

5%    =     5i%0     = 
2%    = 


2i%0 

Argument  —  Since  i%0  cannot  be  taken  from  i%0  one  of  the  5  units  is 
changed  to  20ths  and  added  to  i%0.  This  leaves  4  units. 

Note—  This  last  step  is  difficult  for  children  to  grasp.  The  work 
should  be  developed  very  carefully. 

The  above  example  may  be  presented  as  follows  if  the  teacher  feels 
it  necessary  to  follow  the  general  mechanical  process  used  in  subtrac- 
tion of  whole  numbers. 


*—    2%    = 


Argument  —  Since  i%0  cannot  be  taken  from  i%0  it  is  necessary  to 
add  2%)  to  i%0  making  3%0.  Having  added  one  unit  (2%0)  to  the  minu- 
end we  must  add  one  unit  to  the  subtrahend.  Therefore  we  make  the 
2  a  3. 


=     3i%0 


2% 


4^ 

3% 


EXERCISES. 

5% 
3% 


143/4 

57/8 


25% 


iiimmiiiiiiiimiiiimmmiiiiimmiiiiiniiimiiiiiiiiiiiiiiiimiiin  iiiiniiiii 

MULTIPLICATION. 

1.  When  X  sign  is  read  (times)  the  left  hand  number  is  the  mul- 
tiplier. 

2  X  1/6  is  read  2  times  }£. 

2.  Fractional  multiplicand:  whole  number  multiplier.     Sign  read, 
"times." 

(a)  4  X  ^  =  %  =  2.     Use  objects  or  drawings. 

Argument — 

4  times  1  apple  =  4  apples  J^ 

4  time  1  pencil  =  4  pencils 
4  times  1  half  =  4  halves 

Multiply  numerator  because  it  expresses  the  number  of  parts  used. 

(b)  4   X   %  =  %  =  2% 

Argument — Same  as  above. 

(c)  4  X  %  =  2%  =  10 
Argument — Same  as  above. 

(d)  3  X  2%  =  6% 

Argument — 
2  and  i/5 
3 


6  and  % 

Multiply  the  whole  number  and  fraction  separately. 

(e)     2  X   4%  =  8%  =  9%  =  9i£ 

Argument — Same  as  above. 
94  =  1^8  and  1  are  9 

(e)     Develop  short  cancellation  method — %  x  %  =  2. 
EXERCISES. 


(a) 

(b) 

(c) 

(d) 

2  X  %  = 

2  X  %  = 

3  X  %  = 

2  X   3% 

3  X  %  = 

g      vx   1   /     

5  X  %  = 

3  X   4i/4 

(e) 

2  X   31^  = 

4  X  5%  = 

5X2%  = 

3X2%  = 

3X7%  = 

4X2^  = 

iiiiiiiiiiiiniiiiiiitiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiii tiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiuHiiiiiiiimiiiii 

3.     Whole  number  multiplicand;  fractional  multiplier. 

Recall  the  fact  that  the  product  is  the  same  if  factors  are  inter- 
changed. 

2X3  =  3X2 

Except  in  very  simple  cases  interchange  the  multiplicand  and  mul- 
•tiplier  and  proceed  as  in  step  1. 

In  such  cases  as  %  X  4  the  plan  should  be  to  show  that  ^  X  4  =  % 
of  4. 

After  this  work  has  been  developed  carefully  use  the  short  cancel- 
lation method,  ^  J|  4  =  %  =  2. 

Argument — If  the  teacher  desires  to  use  a  special  plan  for  this  case 
her  argument  would  be 

(a)  %  X  12  =  %  of  12 
14  of  12  =  3 

%  of  12  =  3X3  =  9 

(b)  %  X  8  =  %  of  8 

1st  plan      J/5  of  8  =  1% 

%  of  8  =  4  X    1%  =  41%  =  6% 
2d  plan      %  X   8  =  %  of  *% 

%    Of    49i    =:    % 

%  of  496  =  4  X  %  =  3%  =  6% 
EXERCISES. 


(a) 
%  X     6  = 
•MX     8  = 
%  X  15  = 

%  X   30  = 

(b) 
%  X     4  = 
%  X  12  = 
'  %  X     9  = 
%  X  18  = 

%     X  7 
ft     X   5 
%i  X   9 
H     X  8 

%  X  13 
%  X     3 
%  X     5 
%  X  10 

iMiiiiiiwHwiimmiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiw 

4.     Fractional  multiplicand;  fractional  multiplier. 

(a)  %  X  %  =  %  of  %  =  % 

Argument — 

This  is  based  on  other  steps,  ^  of  4  fifths  =  2  fifths. 

(b)  %  X  %  ==    %  of  %  =  % 
Argument — 

%  of  %  =  % 

%  of  %  =  2  X  %  =  % 

(c)  After  this  step  has  been  covered  carefully"the  short  method 
should  be  developed.  Although  a  complete  explanation  is  not  desirable 
it  is  possible  to  so  arrange  the  examples  that  the  process  seems  a  rea- 
sonable one. 

i£  x  %  =  2/t  may  be  worked  out  by  the  methods 

^  X  ty  =  ii4  =  %  suggested  and  then  worked  by  rule 

and  the  answers  compared. 

The  pupil  will  observe  that  multiplying  the  numerators  together 
and  multiplying  the  denominators  together  produces  the  correct  an- 
swer. When  using  the  short  method  cancellation  should  be  used. 

If  the  examples  used  in  the  first  two  steps  are  carefully  selected 
the  pupils  will  accept  the  short  method  for  such  an  example  as* 

%     X     % 

EXERCISES, 
(a)  (b)  (c)   short  method 

%    x    %  %    x    %  94    x- 

%    x    %  •%    x    %  %    x 

%    x    %  %    x    %  %    x 


miiiiiimiuiiiHiiimmiiiiiiiiHiiii n niiiiiiimiimn IIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII iiiiiiiiiiniiiii iiiiiiiiiiiiiiiiiiiiiiimiiiiiiiin 

Should  the  teacher  desire  to  avoid  the  short  method  she  may  change 
the  multiplicand  to  an  equivalent  fraction  having  a  numerator  divisable 
by  the  denominator  of  the  multiplier. 

%  X   fy  =  %  of  % 
Since  %  =  % 
Then  %  of  i%!  may  be  written 

ft  of  i%! 

%   Of   1%! 

%  x  #  =  ^x%=  ;?        ?  = 

%  X  %  =  %  X   %  =  ^1   X  %  = 

^4  x  %  =  %  X  %  =  %o  X  %  - 

5  When  the  short  method  is  used  examples  involving  mixed  num- 
bers may  be  arranged  by  changing  the  mixed  numbers  to  improper  frac- 
tions. 

%  X  2%  X  %  =  %  X  1%  X  %  =  % 

EXERCISES. 

HX3^X%=  V4X2^X%  = 

1%  X  ?|  X  %  =  %  X  3%  X  %  = 

41  X  2  X  %  =  *  X  1%  X  %  - 

%  X  3%  X  %  =  4V2  X   2      X  %  = 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiim 

DIVISION. 

1.  Define  sign  -r-  to  read,  "divided  by." 

2.  Divisor  a  whole  number;  Dividend  a  fraction. 

(a)  %  -r-  2  =  ^5        Use  drawings  or  objects. 
Argument — 

4  fifths  ~  2  =  2  fifths. 
The  lead  may  be 

4  apples  -4-2  =  2  apples 

4  pencils  -=-2  =  2  pencils,  etc. 
Divide  numerator  because  it  shows  parts  used. 

(b)  9.3  -T-  3  =  3,3        Use  drawings  or  objects. 
Argument — Same  as  above. 

(c)  4%  -*-  2  =  21,4 

Argument — Divide  whole  number  and  fraction  separately. 
2  |  4  and  % 

2  and  i/3  =  2i/3 

If  desired  the  mixed  number  may  be  changed  to  an  improper  frac- 
tion. 

yfo/         .        O     tA/        .        O     T/  Ol  / 

4%  -.-  2  =    1%  -f-  2  ==  %  =  2% 

This  plan  is  best  when  the  example  is  like  5%  -:-  2. 

5%  -*-  2  =  2^  _*.  2  =  1%  =  2ft 

Note — In  some  developments  such  examples  as  %  -r-  3  are  not  dis- 
cussed until  later.    The  development  may  show  that 

5/1  -5-  3  =  i%!  -5-  3  =  %!  if  desired. 

EXERCISES. 

(a)  (b)  (c) 

%  -*-  3  =  %  -*-  3  ==  3%  -s-  2  = 

7/8  -*-  2  =  %  -H  4  =  5%  -4-  3  == 

^--3=  %-3=  6%-J-2  = 

4      -s-  3  = 


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3.     Show  that  6  -r-  2  =  ^  of  6. 

By  induction  show  that  dividing  by  a  number  is  equivalent  to  multi- 
plying by  its  reciprocal.  This  may  be  done  by  using  drawings  and  ob- 
jects. 

(a)  6  -*-  %  =  6  X  %  =  12 
8  -4-  %  =  8  X  %  =  24 

4  -*-  %  =  4  X  %  =  1%  =  6 

(b)  %  -     %  =  %  X  %  =  % 

%  -5-  2      =  $3  X  H  =  %o  =  % 

Mixed  numbers  are  changed  to  improper  fractions,  and  1  is  placed 
under  each  whole  number.  Use  cancellation  when  possible. 

(c)  %  -4-  4      =  %  -s-  ft 


EXERCISES. 
(a)  (b) 


8  -  H  =  ?  fi  -H  2  = 

12  -  y4  =  ?  %  -5-  5  = 

4  -*-  %  =  ? 

6  -  =  ? 


iiiiniiiiiiiiimmiiiiiiimiiiiiiimwwHHiiiiiiimiiiimiiiiiiiu 

4.    If  the  teacher  does  not  use  the  short  method  she  may  change 

fractions  to  equivalent  fractions  having  like  denominators. 

(a)  %  -*-  %  =  2 
Argument  — 

4  apples  -f-  2  apples  =  2 
4  cents  -f-  2  cents  =  2 
4  fifths  -T-  2  fifths  =  2 
Use  the  idea  of  division  by  measurement. 

(b)  4  -s-  fc 

%  -*-  %  =  8 
Argument  —  Same  as  above. 

(c)  %  •*-  % 

%  •*•  10/l5  =  %  =  l%o  =  1% 
Argument  —  Same  as  above. 

(d)  %  H-  5 

%  •*•  2%  =  %o 

EXERCISES. 
(a)  (b)  (c)  (d) 


-    2 


iiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiwim 

FRACTIONAL  PROBLEMS. 
1.     To  find  a  fractional  part  of  a  number. 

(a)  %  of  36 

y4  of  36  =  9 

%  of  36  =  3   X   9  =  27 

(b)  %  of  14 

%  of  14  =  4% 

%  of  14  =  2  X   4%  =  8%  =  9i£ 

(c)  %  of  3fc 

%  of  %  =  %  of  3%0 
%  of  3%o  =  %o 

%  Of  3%0   =  2    X    7^Q    =    1^0    «    1^Q    =    1% 

By  the  short  method. 

(a)  %  of  36  =  %  X   3%'=  27 

(b)  %  of  14  =  %  X  i%  =  2%  =  9% 

(c)  %  of  3%  =  %  X   7^  =  i^0 

EXERCISES. 

(a)  (b)                                     (C) 

%  of  24  =  %  of    7  =  fc  of  2% 

%  of  30  =  %  of  13  =  %  of  3}| 

^  of  28  -  ^  of  25  =  -%  of  4$ 

%  of  20  =  %  of  19  =  ^  of  6% 


IIIIMIIIIIIllllllHIIIMtllllllltlllllllllllllllllllllllllllllllllllltllimillllllllllllllllllllllllllllUIIIIIIIIIIIIIIIIIIU 

2.    To  find  a  number  when  a  fractional  part  of  it  is  known. 

(a)  %  of  a  number  =  8 
^  of  a  number  =M 

%  of  a  number  =  3   X  4  =  12 

(b)  %  of  a  number  ==  8 
y4  of  a  number  ==  2% 

%  of  a  number  =  4  x  2%  =  8%  =  10% 

(c)  %  of  a  number  =  3}£ 
$5  of  a  number  =    19^ 
}$  of  a  number  =    % 

%  of  a  number  =    *%  =  8^ 

By  the  short  method,  sign  X  is  read  "multiplied  by." 

(a)  8  -f-  %  =  8  X  %  =  2%  =  12 

(b)  g  -*-  %  =  8  X   %  =  «%  =  10% 

(c)  3^  -^  %  =  1%  X  %  =  so/6  =  8%  -  8^ 

EXERCISES. 

(a)     Find  number  if        (b)     Find  number  if  (c)     Find  number  if 

%  of  a  number  =  9        %  of  a  number  =  7  %  of  a  number  =  2^ 

%  of  a  number  =  8        %  of  a  number  =  5  %  of  a  number  =  5^ 

%  of  a  number  =  6        %  of  a  number  =  6  ^  of  a  number  =  4% 


To  find  the  fractional  relationship  between  two  numbers. 

(a)  4  is  what  part  of  20? 

Since  there  are  5  4's  in  20,4  is  \'-y  of  20. 
This  is  expressed  by  the  fraction  %0  =  };-, 

(b)  By  analogy. 

5  is  what  part  of  7? 
5  is        of  7 


EXERCISES. 


(a) 

3  is  what  part  of  18 

4  is  what  part  of  24 

7  is  what  part  of  28 

8  is  what  part  of  64 
6  is  what  part  of  36 


(b) 

3  is  what  part  of     5 
2  is  what  part  of    7 

4  is  what  part  of  21 
6  is  what  part  of  11 

5  is  what  part  of  12 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIM 

DECIMAL  FORMS 

1.  Show  that  a  period  may  be  placed  at  the  end  of  a  number. 
264. 

2.  Develop  the  idea  that  fractions  having  denominators  of  10,  100, 
1000,  etc.,  may  be  expressed  in  the  decimal  form. 

%o  =  -3         4/io  =  -4  7/ioo  =  -07         28/100  =  .28     etc. 

3.  Show  that  some  fractions  may  be  changed  to  equivalent  forms 
having  10,  100,  1000,  etc.,  for  denominators. 

%  =  5/io  %o  =  15/ioo 

%  =  tfo    etc. 

4.  Express  the  aliquot  parts  of  100. 

etc. 

EXERCISES. 

392  25  1064 

tio  =  ?  5/io  =  ?  5/ioo  =  ?  32/ioo  =  ? 


% 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 


5.     Show  that  fractions  may  be  expressed  as  equivalent  fractions 
having  10,  100,  1000,  etc.,  as  denominators. 

%!   Of   1^91oO   ==    27%!    ==    '27%! 

IW 

Argument — 

i/ii  of  100  =    91/n 

%!  of  100  =  273/!! 
Note — Short  method  developed  later. 

EXERCISES. 

tt  %  %  %  %  %  % 


iimiiiiiiiimiiiiiiiiimimiiiiiiiiMiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiim 

6.     Develop  the  method  of  multiplying  and  dividing  decimal  frac- 
tions by  10,  100,  etc.,  by  moving  the  point. 

.3  -5-  10  =  .03 
.3  x   10  =  3. 

21.62  -=-  10  =  2.162 

21.62   X  100  =  2162.     etc. 

EXERCISES. 

.5  -f-  10  =  ? 

A  -5-  10  =  ?  .5  X'  10  —  ? 

.4   X   10  =   ?  42.65  -r-  100  =   ? 

32.54  -5-  10  ==  ?  6.345   X   1000  =   ? 

32.54   X   100  =  ?  34.6  -=-  100  =  ? 


Illllllllllltllllillllllllllllllll lllllinilllllllllllllllllllllllHIUIIIIIIIIIIIIIIHIlllHNIIIIIIMIIIIIIIIIIIIIIIIinilliinililllUii.ilillllllillllllllllllHIIIIIIIUIIIIIIIIhllll 

7.     Show  that  to  add  and  subtract  decimal  expressions  the  points 
must  be  placed  one  under  another. 

.12  21.6 

.03  1.08 

.15  22.68 

EXERCISES. 

.25  .32  27.8  63.4  2.54 

04  .01  2.09  2.5  6.34 


Illllllllllllllll  .....  tlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllltll  ......  Illllllllllltllllllllllllltlllllllllllllllllllllllllllllllllllll 

8.     To  multiply  decimal  expressions. 

(a)     21.6 
3 

64.8 
Argument  —  The  carry  may  be  shown  by  writing  the  example. 


63is/10  =  64%0  =  64.8 


(b)         251 
.23 


753 
502 

57.73 
Argument — 


.23    X    251   fc=   -%)0   X   251 

V100  of  251  =  2.51 

2%00  of  251  =  23   X   2.51  =  57.73 


(c) 


103-32 
Argument — 

16.4  =  6   X    16.4  and  V10    X    16.4 
6.3 

16.4 

6  3/10   X   16.4  =  1.64 

.y10   X   16.4  =  3   X    1.64  =  4.92 
98.4 

98.4 
4.92 


103.32 

EXERCISES. 


26.4 
8 

264 
.3 

2.56 
4 

634 
.25 

.36 
4 

84.6 

2 

423 
3.6 

265 
1.26 

6.8 
1.6 

25.6 
.14 

6.23 
.03 

.634 
2.5 

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9.     To  divide  decimal  expressions. 

(a)  2.31 

2  |  4.62 


(b)      .03  |  369 

Multiply  both  dividend  and  divisor  by  100 
12300 


3  |  36900 

Argument-— Show  that  dividend  and  divisor  may  be  multiplied  by 
same  number  without  changing  their  relationship. 


(c)     2.34  ]  64.6 
Multiply  both  dividend  and  divisor  by  100 

27.606  + 
234  I  6460.000 
468 

1780 
1638 


1420 
1404 


1600 
1404 


Argument — Same  as  above. 

Note — It  is  very  important  to  perfect  the  mechanical  process. 

1.  Multiply  both  dividend  and  divisor  by  10,  100,  etc.,  so  that  the 
divisor  is  made  a  whole  number. 

2.  Place  a  point  in  the  quotient  directly  over  the  new  position  of 
the  point  in  the  dividend. 

3.  Place  each  digit  of  the  quotient  directly  over  the  last  digit  used 
in  the  dividend. 

EXERCISES. 


3  |  64.3  .6  |  256  2.5  |  63.4 


5  |    .634  1.6  1  463  .43  |  624. 


2  [  4.36  .04  |  634  .24  |  1.634 


II IIIIIIIINIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIN 

10.     Develop  the  short  method  for  changing  a  fraction  to  a  decimal 
form. 

.2787  + 


3/n     11  |  3.000 
22 

SO 

77 


30 
22 

80 

77 

3 

EXERCISES. 

% 


I Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIIMIIIHIIIIIIMIIIIII 

11.     The  problems  of  decimal  fractions: 

(a)  To  find   .08  of  64 
ilnu  of  64  =    .64 

SIIM)  of  64  =  8   X    .64  =  5.12 

(b)  To  find  the  number  if  .08  of  it  is  64 
s|(l()  of  number  =  64 

'  ]uo  of  number  =  8 

io<)100  of  number  =  100   X   8  =  800 

(c)  To  show  relationship  between  .08  and  64 
.08 

fractional  relationship 
64 

.00125 
.08    =  =    64  fTOSOOiT 

64 
64 

160 
128 

320 
320 


Argument — A  fraction  represents  a  division. 
Short  methods: 

(a)      .08  of  64  =       64 

.08 


5.12 

800 


(b)  64  ~-   .08  =   .08  |  64  =  8  |  6400 

.00125 

(c)  .08  =    64  I   .08000 

64 
64 

160 
128 

320 
320 

EXERCISES. 

Find  Find  Number  If  Find  Relationship  Between 

.06  of  48  .03  of  it  =  18                     .03  and     1.4 

.12  of  72  .45  of  it  =  90                      25  and  1.06 

.05  of  27  .15  of  it  =  75                    1.6  and  24.6 


iiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiimiiiiimiiiiiiiiiNH 

PERCENTAGE. 

1.  To  define  %  sign.    Word  percent. 

2%  =  %oo  =  -02 

2.  To  show  that  fractions  having  denominators  of  100  may  be  ex- 
pressed by  using  %  sign. 

(a)  Moo  =3% 
4y100  =  41% 

(b)  .04  =     4% 
.16  =  16% 

3.  Drill  on  aliquot  parts  of  100 

4.  Express  fractions  in  percents 

(a)  %!  =  %!  of  ioo/100  =  27%n  hundred ths  =  27§/n% 

(b)  Short  Method 

.27  + 

3/n  =  11  I  3.00 
22 

80 

77 

3 

Argument — A  fraction  represents  a  division. 

(c)  %!  of  100% 

Mi  of  100%  =  9Mi% 

3/n  of  100%  =  3   X   9i/n%  = 

EXERCISES. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiimiiiiiiiiiiimiiiiiiM 

5.     To  find  a  percent  of  a  number. 

(a)  5%  of  86  =  %0o  of  86 
y100  of  86  =   .86 

%00  of  86  =  5   X    -86  =  4.30 

(b)  Short  method 

86 
.05 

4.30 

EXERCISES. 

3%  of  15  6%  of  19  3%  of  142 

1%  of  28  4%  of  35  6%  of  263 

8%  of  72  15%  of  96  15%  of  128 

12%  of  48  26%  of  172  49%  of  642 


Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Illlllllllllll Illllllllllllllllllllllllllllllllllllllllllllllllllllllll 

6.     To  find  a  number  when  a  percent  is  known. 

(a)  86  =  5%   of  what  number? 
5%  =  86 

1%   =  17.2 
100%  =  1720 

(b)  Short  Method 

1720. 


.05  j  86  =  5  |  8600. 

EXERCISES. 

Find  Number  If 

8%   of  it  =  16  5%   of  if  =  14 

10%  of  it  =  50  8%  of  it  =  27 

5%  of  it  =  35  14%   of  it  =  46 

6%  of  it  =  66  70  %  of  it  =  172 


iiiiiiiiiiiiiiiminiHHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiM 

7.     To  find  what  percent  one  number  is  of  another 

(a)  8  is  what  %   of  86 

8£,6  =^=  fractional  relationship 

%6  of  100% 

]£e  of  100%  =  1.1627% 

%Q  of  100%  =  8   X   1.162  =  9.3016% 

(b)  Short  Method 

.093  + 
86  !  8.000 

774 

260 
258 


2 

093  =  9.3% 


EXERCISES. 


Find  What  % 

5  is  of  25  14  is  of  28 
10  is  of  60  8  is  of  13 

8  is  of  48  7  is  of  43 

6  is  of  90  56  is  of  362 


illlllllllllllllllllllllllllim IIIIIIIIIIHIIIHIIIIIIIIIIIIHIIIIIIHIIIIIHIIIIIIIIHIIIIIIIIIIIIIItllllMllinillllllllllllllllllllllltlllHIIIIIIIIIIIHIIIIIIIIIIIIIIIIHIIIIIIIIfl 


SPECIAL  TOPICS 

Under  the  plan  of  accepted  courses  of  study  the  pupils  are  to  con* 
plete  the  work  devoted  to  the  common  operations  in  whole  numbers, 
common  fractions,  decimal  fractions,  and  percentage  in  the  first  six 
years.  During  this  time  they  are  to  study  the  common  measuring 
units,  their  relationships,  and  perform  all  the  ordinary  measurements. 

The  next  logical  course  is  one  given  over  to  the  common  appli- 
cations of  facts  and  processes,  and  the  ordinary  measurements.  The 
dominant  feature  of  the  work  should  be  the  composite  problems  in- 
volving the  application  of  many  topics.  An  example  of  such  a  prob- 
lem is  the  building  of  a  house.  Pupils  study  the  smaller  problems 
involved,  the  business  problems  connected  with  the  buying  of  the  lot, 
the  measurements  involved  in  the  cellar  excavation,  the  building  of  the 
wall,  the  labor  costs,  plastering,  papering,  painting,  roofing,  furnish- 
ing, and  finally  the  business  problems  connected  with  selling  and  the 
housewives'  problems  connected  with  living-  in  the  house. 

These  composite  problems  should  take  the  forms  of: 

Buying  and  selling  problems 
Construction  problems 
Communication  problems 
Transportation  problems 
Money  and  credit  problems 
Production  problems 
Etc. 

In  all  such  problems  the  center  of  interest  should  be  on  the  ap- 
plication rather  than  on  topics.  The  problems  should  draw  from  many 
topics.  For  example  a  problem  in  buying  and  selling  must  of  neces- 
sity touch  upon  communication,  transportation  and  money  and  credit. 
A  construction  problem  may  deal  also  with  buying  and  selling,  com- 
munication, transportation,  money  and  credit  and  possibly  production. 

A  knowledge  of  general  business  procedure  is  needed  by  every- 
one regardless  of  special  vocation  or  profession.  The  problems  con- 
nected with  banking  are  those  which  we  are  interested  in;  how  to 
open  an  account,  how  to  write  a  check,  how  to  open  a  savings  bank 
account,  etc.  The  problems  connected  with  taxes,  insurance,  bonds 
(such  as  liberty  bonds)  corporations,  investments,  etc.,  are  interesting 
to  more  and  more  people.  The  war  brought  about  much  general  edu- 
cation along  these  lines.  A  course  of  this  sort  is  needed  and  it  must 
be  entirely  different  from  the  old  topical  courses  offered  in  the  eighth 
grade  work.  It  must  be  given  in  connection  with  actual  business  pro- 
cedure. 

Modern  business  is  so  interlocking  no  person  can  carry  on  the 
ordinary  activities  of  social  life  without  knowledge  of  common  busi- 
ness practice.  It  is  not  necessary  or  desirable  to  present  topics  as 
completely  as  has  often  been  done  in  the  past.  The  special  technical 
instruction  in  any  branch  of  business  can  better  be  given  to  those  par- 
ticular students  who  enter  it  as  a  life  occupation  by  those  in  charge  of 
the  special  branch. 

For  example  a  study  of  all  the  highly  technical  problems  of  in- 
surance is  of  little  interest  to  society  at  large  but  all  people  should 
know  the  values  of  insurance,  the  kinds  of  insurance,  how  to  select 
a  policy,  how  to  take  out  a  policy,  how  to  pay  premiums  and  how  to 
collect  insurance.  Or  again  in  taxation  the  problems  should  cover  the 
needs  for  taxes,  the  meaning  of  a  tax  rate,  the  kinds  of  taxes,  how  the 
tax  rate  is  established,  how  to  pay  taxes,  the  penalties  for  not  paying 
taxes,  the  value  of  a  tax  receipt,  tax  sales,  how  to  make  out  an  in- 
come tax  form,  etc. 


Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Ill I Illllllllllllllllllllllllllllllllllllllllllllllllllfllfllllllllllllllllllllllllllllllllll 

After  a  course  in  which  pupils  have  selected  and  applied  the 
arithmetical  knowledge  they  have,  the  classification  of  arithmetical 
knowledge  in  terms  of  local  needs  would  seem  advisable.  For  ex- 
ample there  is  a  certain  group  of  arithmetical  facts  and  applications 
of  interest  especially  to  farmers  and  another  group  of  facts  and  ap- 
plications of  especial  interest  to  those  who  live  in  cities.  Pupils  should 
study  the  arithmetical  needs  of  society  in  terms  of  the  needs  of  the 
larger  groups. 

In  this  work  attention  should  be  given  to  the  significance  of  so- 
cial problems  as  well  as  to  individual  ones.  These  problems  differ 
from  those  of  the  individual  in  that  they  are  connected  with  social 
welfare  rather  than  individual  welfare.  For  example  when  studying 
the  relation  of  mathematics  to  rural  life  the  student  is  interested  not 
only  in  the  problems  of  the  farm  but  those  of  rural  community  life. 
A  like  plan  should  be  followed  when  studying  the  relation  of  mathe- 
matics to  urban  life.  There  are  problems  of  the  home  and  problems 
of  the  community  at  large. 

In  the  construction  work  the  students  cover  the  definition,  classi- 
fication, and  mensuration  of  the  common  geometrical  figures.  This 
subject  matter  in  connection  with  applied  problems  helps  the  students 
to  acquire  a  fund  of  useful  knowledge  relating  to  the  mathematics  of 
form. 

After  the  general  business  applications  are  discussed  the  special 
mathematical  applications  needed  in  vocational  lines  should  be  taken 
up.  Mathematics  of  wood-working,  of  farming,  of  shop  practice,  of 
automobile  work,  of  cement  work,  of  clerking,  of  dress-making,  etc., 
should  be  taken  up  with  the  thought  that  boys  and  girls  may  get  a 
spark  of  inspiration  somewhere  along  the  line.  In  some  of  the  voca- 
tions they  will  find  an  interest. 

This  course  may  seem  to  duplicate  some  of  the  subject  matter 
but  the  stress  is  placed  on  the  vocation.  The  thought  is  that  stud 
ents  should  receive  special  preparation  for  entering  some  particular 
occupation.  It  is  to  be  hoped  that  individual  students  will  become 
especially  interested  in  some  particular  phase  of  this  presentation. 

In  case  they  do  this  the  teacher  should  encourage  them  to  study 
appropriate  problems.  A  well  arranged  reference  library  is  very  val- 
uable for  use  in  this  grade.  There  should  be  books  in  such  a  library 
on  the  mathematics  of  farming,  retail  trade,  commerce,  industry,  clerk- 
ing, trades,  salesmanship,  construction,  etc.  Class  work  should  be  sup- 
plemented by  talks  by  persons  in  the  community  engaged  in  various 
occupations  and  by  visits  to  business  houses.  If  possible,  arrangements 
should  be  made  to  allow  students  to  engage  in  the  actual  work  they  feel 
interested  in. 

Note:— The  following  outlines  are  arranged  to  suggest  the  import- 
ant points  to  develop  under  various  topical  headings.  No  attempt  has 
been  made  to  arrange  complete  topical  outlines  or  to  present  a  substi- 
tute for  a  text  book. 

In  upper  grade  work  a  text  book  together  with  reference  books 
should  always  be  used.  The  teacher  should  select  and  stress  the  sub- 
ject matter  that  is  most  closely  connected  with  life  problems  and 
should  present  this  subject  matter  in  such  a  way  as  to  appeal  to  the 
interests  of  students. 

The  following  two  outlines  on  communication  and  transportation 
suggest  the  general  plan  to  be  followed  when  taking  up  life  activities. 


iiimimiimniiiimiiiiiiiimiiiiiimiiimiiiiiiiimimiiimiiiiiH  

TRANSPORTATION 

1.  Railroad 

Freight 

Classification 
Rates 

Freight  bills 
How  to  send 
How  to  receive 

Express 

Classification 
Rates 
Receipts 
How  to  send 
How  to  receive 

Parcel  Post 

Classification 

Rates 

Limits  of  weight 

Insurance 

How  to  send 

How  to  receive 

2.  Shipping  by  boat 

Classification 

Rates 

Bills  of  shipment 

How  to  send 

How  to  receive 

3.  Shipping  by  motor  truck 

Rates 

How  to  send 

How  to  receive 

4.  Air  transportation 

Values 

Discuss  development 

5.  Discuss  all  above  tonics  with  regard  for  values  and  probable 
development.    Work  out  problems  dealing  with  each  point. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIII Illlllltlll Illlllllllllllllllllllllllllllllllllllllll 


COMMUNICATION 

Postal  service 

Classification  1st,  2nd,  3rd  class,  etc. 
Limit  of  weight — Regulations  for  sealing,  etc. 
Rates  on  different  classes. 
Exceptions, — U.  S.  bulletin,  etc. 
Foreign  postage 
Registered  mail. 
Special  delivery. 

Discuss  all  above  points  and  work  out  problems  dealing  with 
each. 

Telegraph  and  cable 
Classification — day  and  night  messages. 

day  and  night  letters. 
Rates 

Cable  rates 

Advantages  of  telegraph 
How  to  send  message 
How  to  receive  message 

Wireless 

Uses 

Discussion  covering  its  development. 

Discussion  covering  equipment  needed. 

How  to  send  and  receive  messages. 

Air  Postal  service 
Uses 

Discussion  covering  its  development 
Rates 
How  to  send  and  receive  messages. 

Telephone — Local  and  long  distance. 

Uses,  rates,  etc.    How  to  send  and  receive  messages. 


iiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiMN 

COMMON  BUSINESS   PAPERS 

1.  Discussion  covering. 

Receipt 

Order. 

Statement 

Receipted  bills 

Account 

Ledger 

Day  book 

Contract 

Freight  bill 

Etc. 

2.  Have  all  such  papers  for  inspection  and  have  pupils  fill  out 
blank  forms. 

3.  Discuss  common  words  and  their  abbreviations 

4.  Study  the  form  of  keeping  an  account  and  have  pupils  keep 
an  account  of  simple  transactions. 


Illlllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllltllllllllllllllllllllllllllllllllllllllllllllllllllll Illllllllllllll Illllllllllll 

COMMISSION— BROKERAGE 

1.  Definition. 

2.  Uses. 

3.  Definition  of  Terms. 

The  Commission. 

Rate. 

Net  Proceeds. 

Principal. 

Agent. 

Broker. 

Etc. 

4.  Problems  in  Commission. 

A.     To  find  a  commission. 

To  find  5%  commission  on  sales  amounting  to  $4,684.62: 


First  Method— 

100%  =  $4684.62 
1%  =  46.8462 
5%  =  $234.231 


Second  Method — 

$4684.62 

.05 


$234.2310 


B.  To   find   principal   when   the  rate   and    commission    are 
known : 

6%  commission  on  sales  yields  $64.72.  Find  the  amount  of  sales. 
6%  =  $64.72 

1%  =  $10.78%  $1078.66% 

100%  =  $1078.66%  .06  |  064.7200 

C.  To  find  rate  when. the  principal  and  the  commission  are 
known : 


Principal  =  $643.70 


Commission  =  $172.63 
$64; 


.268  + 


172.63 


The  relationship  = 


172.63 


643.70 


X   100% 


$17263.000 
128740 

438900 
386220 


=  26.81   +   % 


526800 
514960 


iiiiiiiiiiiiini iiiiimiiiiiiiimiiiiiiiiiiiiiiiiiiimiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiH 

COMMERCIAL  DISCOUNT 

1      Definition. 
) 

2.  Uses. 

3.  Definition  of  Terms. 

Marked  Price. 
Net  Price. 
Rate  of  Discount. 
Successive  Discounts. 
Trade  Discount. 
Time  Discount. 
Cash  Discount. 
Etc. 

4.  Problems  in  Discount 

A.  To  find  a  discount  when  marked  pric'e  and  rate  are  given: 

Marked  price  =  $762.14 
Rate  =  6% 

First  Method—  Second  Method— 

100%  =  $762.14  $762.14 

1%'=  $7.6214  .06 

6%  =  $45.7284 

$45.7284 
Note— Net  price  =  94%  =  $716.4116 

B.  To  find  marked  price  when  the  rate  and  the  discount  are 
given : 

Discount  =  $17.50         15%  =  discount 

Second  Method — 
$116.66 


.15  |  $17.50 

15 

First  Method—  — - 

15%  =  $17.50  25 

1%  =  $1.1666%  15 

100%  =  $116.66% 

100 
90 

10 


iiiiiiiiiinii iiiiiuii iiiiiiiiiiiiiiiiMiiiini!iiiiiiiiiiiuiiiiniiiiuiiiiiiiiiiiiuiiiiiiiiiiiininiiiiiiiiiiiMiMiiiiiiii!iiiiiiiiMiiiiiiiiiiiiiiiniiiiiiiiininiMiitiiin 


C.     To  find  the  rate  when  the  marked  price  and  discount 
are  given: 


Marked  price  =  $28.76         Discount  = 
4.36 


.151  + 


28.76  |  $4.36000 
2876 


The  relationship  =  - 


436.% 


4.36 

-  x  100%  = 

28.76  28.76 


28.76 


=  15.12% 


D.     To  find  successive  discounts: 
Marked  price  =  $462.14        Successive  discounts 


14840 
14380 

4600 
2826 

1724 
20%,  10% 


First  Method— 

100%  =  $462.14 
1%  =         4.6214 
20%  =         92.428 
80%  =  $369.712 

100%  =  $369.712 

1%  =         3.69712 
10%  =       36.9712 
90%  =  $332.7408 


Second  Method — 
$462.14 
.80 


$369.7120 
.90 

$332.740800 


iiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiijiiiiiiiiiiiiiiiiiiiimu  iiiniii 

INTEREST 

1.  Definition  of  Interest. 

2.  Definition  of  Terms. 

Principal. 

Rate  of  Interest. 

Time. 

The  Interest. 

Amount. 

3.  The  procedure  in  finding  interest. 

First  Plan — Find  interest  on  $150  for  2  years  6  months  at  6%. 
$   .06  interest  on  $1  for  1  year. 
2*/£  X  $   .06  =  $   .15  interest  on  $1  for  2y2  yrs. 
150  X  $   .15  =  $22.50  interest  on  $150  for  2%  yrs. 

Second  Plan — 

$   .06  interest  on  $1  for  1  year. 

150  X  $   .06  =  $9.00  interest  on  $150  for  1  year. 

2%  X  $9.00  =  $22.50  interest  on  $150  for  2%  years. 

SPECIAL  METHODS 
A 

$  .  06  interest  on  $1  for  1  year 

$  .  005  interest  on  $1  for  1  month 

$  .  000%  interest  on  $1  for  1  day 

2  X  $   .06     =  $   .12  interest  on  $1  for  2  years 

6  X   $   .005  =  $   .03  interest  on  $1  for  6  months 


$   .15  interest  on  $1  for  2  years  6  months 
150  X  $   .15  =  22.50  interest  on  $150  for  2  years  6  months 

Note— If  the  rate  is  5%  take  %  o*f  this  answer 
If  the  rate  is  4%  take  %  of  this  answer 

B 

60  days  =  %  of  year 

At  1%  the  interest  on  $150  ==  $1.50  for  1  year 

Since  60  days  =  %  of  year  and  6%  =6  X  1% 

The  interest  on  $150  for  60  days  at  6%  =  $1.50 

There  are  15  60-day  periods  in  2i/>  years 

15   X   $1.50  =  $22.50 

Note — Same  as  above. 

Find  interest  on  $56  for  80  days  at  4% 

$   .56      interest  on  $56  for  60  days  at  6% 

$   .18%  (%  of  $   .56)  interest  on  $56  for  20  days 


$   .  74%  interest  on  $56  for  80  days 

%  of  $   .74%  =  $   .497/6  interest  on  $56  for  80  days  at  4% 

4.     Compound  Interest. 

A.  Definition. 

B.  Method  of  working  out. 

C.  Formula  for  compound  interest. 


A  =  P»CM    Iff-)   P   (1  +   RT)n 

A  =  amount. 

P  =  principal. 

R  =  rate. 

T  =  time  in  years,  between  the  compoundings. 

N  —  number  of  compoundings. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii imiiiiiiiiiiiimi in IIIIIIIIIIMIIIIIIII iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii 

D.     Problems. 

To  find  compound  interest  on  $497.68  for  2  years  at  6%,  interest 
compounded  semi-annually : 

6%  for  1  year  =  3%  for  6  months. 


$497 .1 


$14.9304  interest  for  6  mo. 
$497.68 


$512.6104  amount  after  6  mo. 
.03 


$15.378312  interest  2d  6  mo. 


By  Formula — 

1  +  RT  =  1.03 

(1    X   RT)4  ==  1.125  + 

1.125   X   $497.68  =  $559.88  + 
Note — This  may  be  carried  out  to 

greater    accuracy    by    extending 

1.125  + 


$15.38 
512.61 


$527.99  amount  after  1  year 
.03 


$15.8397  interest  third  6  mo. 
527.99 


$543.8297  amount  after  18  mo. 
.03 


$16.314891  interest  after  2  years 
$543.8297 


amount  after  2  years 


To  find  compound  interest  on  $9040  for  4  years  at  4%,  interest  com- 
ix unified  annually: 

$9040 
.04 


$361.60  Int.  for  1  yr. 
$9040. 


$9401.60  Amt.  after  1  yr. 
.04 


$376.0640  Int.  after  2  yrs. 
$9401.60 


$9777.6640  Amt.  after  2  yrs. 
.04 


$391.106560  Int.  after  3  yrs. 
$9777.66 


$10168.76656     Amt.  after  3  yrs. 
.04 


$406 . 7506624     Int.  after  4  yrs, 
$10168.766 


$10575.516 


Amt.  after  4  yrs. 


By  Formula — 

A  ==  P   (1   +  RT)4 

1   +  RT  =  1.04 

(1  +  RT)4  =  1.1698  4- 

1.1698  4-    X   $9040  =  $10574.99 

Note — This  may  be  carried  out  to 
greater  accuracy  by  extending 
1.1698  + 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM 

BANKING 

1.  A  bank  Provides 

A  place  to  deposit  money  for  safe-keeping 

A  place  to  loan  money  not  needed.   (Saving  Dept.) 

A  place  to  borrow  money. 

A  checking  privilege 

A  way  to  send  money  (bank  draft) 

A  clearing  house  for  checks  on  many  banks,  etc. 

2.  Classification 

National  Banks 
State  Banks. 
Private  Banks. 
Postal  Savings  Banks 
National  Reserve  Banks 
Trust  Companies 
Savings  Banks 

3.  Definition  of  terms. 

Checking  account. 

Savings  account. 

Deposit  slip. 

Check  and  check  book, 

Draft. 

Note. 

Discount  on  note 

Security. 

Safety  deposit  box. 

Indorse 

Overdraft. 

Etc. 

4.  Problems. 

To  open  a  checking  account* 

To  open  a  saving  account 

To  write  a  check. 

To  draw  out  money  from  a  savings  account. 

To  buy  a  bank  draft. 

To  borrow  money. 

To  sell  a  note. 

To   rent  a  safety  deposit  box. 

Note: — All  papers  should  be  made  out  by  pupils. 


IIIJIIIIIIIIIIIIIIIIIIIII Illlllllllllllllllllllllllllll II I Illllllllllllllllllllllllllllllllllllllllllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllll 

PROMISSORY  NOTES 

1.  Definition. 

2.  Essential  points. 

(a)  A  note  should  state  the  time  and  place  when  written. 

(b)  A  note  should  state  the  time  and  place  when  due. 

(c)  A  note  should  state  the  name  of  the  person  to  whom 
payment  is  to  be  made. 

(d)  A  note  should  state  the  name  of  the  person  making  the 
promise  to  pay. 

(e)  A  note   should  usually  state   that  value  has  been  re- 
ceived. 

(f)  A  note  should  state  the  rate  of  interest. 

3.  Definition  of  terms. 

Maker. 

Payee. 

Indorse    i  In  blank)    (In  full)    (In  limited  form). 

Negotiable. 

Maturity. 

4.  Values. 

Provides   a  record. 

May  be  transferred  (sold  to  a  bank  for  example). 

May  be  collected  through  the  court. 

5.  Partial   payment  of  promissory  notes. 

The  United  States  Rule. 

Kind  the  amount  of  the  principal  when  the  first  payment  is  made. 
From  this  amount  subtract  the  first  payment.  The  remainder  is  the 
new  principal.  Continue  until  final  payment  is  made. 

It'  a  payment  is  less  than  the  interest,  the  payment  is  not  sub- 
tracted. It  is  carried  to  the  next  payment.  This  procedure  is  follow- 
ed until  a  total  payment  is  equal  to  or  greater  than  the  interest. 

ILLUSTRATIVE    EXAMPLE 

.  Note  for  $2500.         6%  interest.  Date  July  15,  1919. 

August  15,  1919,   $100  is  paid. 
January  1.  1920,  $10  is  paid. 
June  15,  1920,  $200  is  paid. 
Find  amount  at  this  last  date. 

Interest  from  July  15,  1919  to  August  15,  1919=$12.50 
Amount  of  note  August  15,  1919=$2512.50. 
Amount  of  note  less  $100— $2412.50. 

Interest  on  $2412.50  from  August  15,  '19  to  January  1,  '20=$54.28 
Amount  of  note   January   1,   '20=$2466.78. 
Payment  not  subtracted. 

Interest  on  $2466.78  from  January  1,  '20  to  June  15,  '20=467.83. 
Amount  of  note  June  15,  '20=$2534.61. 
Amount  of  note  less  $210=$2324.61. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1H 

STOCKS. 

1.  Definitions  of  corporations  and  capital  stock. 

2.  Needs  for  corporations. 

A.  Enables  an  organization  to  transact  business  as  an  in- 
dividual. 

B.  Enables  individuals  to  invest  money  without  taking  an 
active  part  in  the  activities  of  the  company. 

C.  Makes  larger  combinations  of  capital  possible,  which  in 
turn  makes  greater  projects  possible. 

D.  Offers  the  person  having  small  capital  an  opportunity  to 
join  in  the  promoting  of  a  large  venture. 

3.  Definition  of  terms: 

Share  of  stock. 
Par  value. 
Market  value. 
Discount. 
Premium. 
Stock  holder. 
Assessment. 
Dividend. 
Stock  Broker. 
Brokerage. 

4.  Ways  to  buy  stocks: 

A.  Direct  buying. 

B.  On  margins. 

C.  Partial  payment  plan. 

5.  Examples: 

A.     Find  the  cost  of  40  shares  of  stock  in  a  stove  factory 
bought  at  91  with  brokerage  at  y8%. 

40   X  $91         =  $3640  cost  of  stock  at  91 

40   X   $    .125  =  5  cost  of  brokerage  at  ys% 


$3645  total  cost  of  stock 

B.     I  must  raise  $950.     How  many  shares  of  stock  at 
must  be  sold  to  meet  this  amount  with  brokerage  at  }&%? 
$88.50     price  of  one  share 
.125  brokerage 

$88.375  net  price  per  share 

10.749  +         No.  shares  of  stock. 

88.375  |~950000T~  (H   shares  since  one  share  cannot 

88375  '  be  divided.) 


662500 
618625 

438750 
353500 

852500 


Illllllllllllllllllllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllllll I Illllllllllllllllllllllllllllllllllll Illlllllllllllllllil Illllllllllllllllll 

C.     Find  the  cost  of  450  shares  of  stock  (par  value  100)  at 
3%  premium,  brokerage  %%. 

$100.         cost  of  each  share  at  par 
3 .         premium 
.125  brokerage  on  each  share 


$103.125  total  cost  of  each  share 
450   X   $103.125  =  $46406.25.     Cost  of  450  shares 

D.  I  buy  60  shares  of  stock  (100  par  value)  for  88  If  the 
stock  pays  6%  on  par  value  what  is  the  rate  at  88?  What  is  the 
total  interest? 

60   X  $6  =  $360.    Total  interest. 

$88!   =  cost  of  each  share 
6.   ==  interest  on  each  share 

.0681  rate  at  88 


88  |  6.00 
5  28 


720 
704 

160 

E.     A  share  of  stock  yields  15%  when  purchased  at  70  (par 
value  100).    What  rate  is  this  equivalent  to  on  par  value? 

15%  at  70  yields  $10.50. 

.105  rate  on  par  value 

100  |  1~050 
100 

500 
500 


iiiiiiiiiiiiiiiiiiiiiiiiiimiiriiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiini iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii iiiiiiiiniiiiiiiii 

BONDS. 

1.  Definition. 

2.  Relation  of  a  bond  to  a  promissory  note. 

3.  Relation  of  bonds  to  shares  of  stock. 

4.  Classification. 

U.  S.  Government. 

State. 

City. 

County. 

Industrial. 

Special. 

School. 

Bridge. 

Library. 

Paving. 

Etc. 

5.  Definition  of  terms. 

Mortgage. 

Premium. 

Discount. 

Coupon  bond. 

Registered. 

Quotation. 

Bond  holder. 

Commission. 

Brokerage. 

Par  value. 

Market  value. 

6.  How  to  buy  bonds. 

7.  Examples: 

A.  Find  cost  of  25  Liberty  Bonds  4}4%  at  92,  brokerage  1  s- 
$   .125  brokerage  on  one  bond. 

25  X  $   .125  =  $3.125  brokerage  on  25  bonds. 

25  X  $92  =  $2300  cost  of  25  bonds  at  92. 

$2300  +  $3.125  =  $2303.125  total  cost  of  25  bonds. 

B.  Find  par  value  of  4%  city  bonds  yielding  $1200. 

$30000.  par  value  of  bonds. 
04.  7Tl200007~ 

C.  How  much  must  be  invested  in  bonds  at  102  to  yield  $600 
per  year  at  4i/,%,  brokerage  1 8. 

At  4}£%  it  would  take  $13333.33  -f  to  yield  $600. 
Or  134  $100  bonds. 

$13400         cost  of  bonds  at  par. 
268         $2  premium  on  each. 
16.75  brokerage. 


$13684.75  total  cost  of  investment 


iiiiiiiiiiiiiiimiiiiiiiiiiiiiniiiiii iiMiiiiiiiiiiiuniiiiiiiiiiiiiiiiiniiiiiiiiiHiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniin 

D.     If  bonds  yield  4y2%    (100  par  value)   what  percent  will 
they  yield  if  they  can  be  bought  for  90? 
$90  investment. 
$4.50  interest  on  each  bond. 

.05  rate  of  interest. 


90  I  4.50 
4  50 


E.     A  bond  is  at  32%  premium,  what  rate  does  it  pay  if  it 
bears  4U%  (par  100)  ? 

$132  cost  of  bond. 
$4.50  interest  on  bond. 

.0340  +  rate  of  interest. 
132  |~4750~ 
396 

540 

528 

120 


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TAXES— DUTIES— INTERNAL  REVENUES. 

1.  Definition. 

2.  Needs  and  uses  of  taxes,  duties  and  revenues. 

To  pay  expenses  of  government. 

To  pay  for  improvement  of  property. 

To  buy  property. 

To  pay  for  protection,  police,  fire,  accident,  etc. 

To  pay  for  education  of  children. 

To  pay  for  care  of  insane,  disabled  and  poor. 

To  pay  for  care  of  criminals. 

To  pay  for  care  of  roads  and  new  roads. 

To  pay  for  enforcement  o.f  special  laws,  prohibition,  etc. 

To  pay  for  harbors,  dredging,  canals,  etc. 

To  pay  for  conservation  of  natural  resources,  forestry,  etc. 

To  pay  interest  on  old  debts,  war,  etc. 

To  pay  army,  navy,  etc. 

'.  Note — This  list  may  be  extended  by  students. 

3.  Classification. 

Real  estate. 
Personal  property. 
Poll. 
School. 

Manufactured  articles. 
Imported  articles. 
Special  taxes — 

.Automobiles. 

Hunting  and  fishing. 

Income. 

Inheritance,  etc. 

War  times,  etc. 

4.  Definition  of  terms. 

Assessor. 

Collector. 

Tariff. 

Ports  of  entry. 

Free  list. 

Ad  valorem  duty. 

Invoice. 

Specific  duty. 

Tare. 

Assessed  valuation. 

Rate. 

Tax. 

5.  Problems  in  taxes. 

A.     To  find  the  tax  when  the  assessed  valuation  and  the  rate 
are  given. 

$4,500,000 — Assess-ed  valuation  of  city  property. 
13  mills  on  the  dollar— rate. 
$4,500,000 
.013 


13500000 
4500000 

$58,500.000  tax 


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B.     To  find  the  rate  when  the  assessed  valuation  and  the  tax 
are  given. 

$8000  =  assessed  valuation. 
$72  =  tax. 

.009  rate 
8000  |  72.000 
72  000 


C.     To  find  the  assessed  valuation  when  the  rate  and  the  tax 
are  given. 

$156— tax. 
.  014— rate. 

$11142.857  +  assessed  valuation. 


Vpj.4.1  156000 
14 

16 
14 

20 
14 

60 
56 

40 

28 

120 
112 

80 
70 

100 
98 

2 

6.  Method  of  assessing  property. 

7.  Method  of  establishing  rate. 

$456,000— assessed  valuation. 
$26000— tax  needed. 

.057  +  rate 


456,000  |  26000.000 
2280000 

3200000 
3192000 


8000 
Method  of  paying  taxes  (value  of  tax  receipt,  etc  ) 


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INSURANCE. 

1.  Definition. 

2.  Values.  \ 

A.  Protection. 

B.  Savings.  "> 

C.  Investment. 

3.  Classification. 

A.  Property  insurance. 

Fire. 

Tornado. 

Plate  glass. 

Flood. 

Marine. 

Etc. 

B.  Personal  insurance. 

straight  life  policy. 
Life  limited  payment  policy. 

endowment  policy. 
Accident. 
Sickness. 

C.  Other  types. 

Insurance  against  rain. 
Insurance  against  frost. 
Etc. 

4.  Definition  of  terms. 

Policy. 

Premium. 

Rate. 

Paid  up  insurance. 

Endowment. 

Dividend. 

Loan  on  policy. 

Etc. 

5.  Taking  out  insurance. 

6.  Payment  of  premiums. 

7.  Method  for  collecting  insurance. 

8.  Method  for  borrowing  money  on  policy. 


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LONGITUDE  AND  TIME. 

1.  Definitions. 

Meridfen. 
Equator. 
Prime  Meridran. 
Longitude. 

2.  Relation  between  360°  of  longitude  and  24  hour  day. 

15°  longitude  corresponds  to  1  hour  of  time. 
15'   longitude  corresponds  to  1  minute  of  time. 
15"  longitude  corresponds  to  1  second  of  time. 

4  minutes  of  time  corresponds  to  1°  longitude. 

4  seconds  of  time  corresponds  to  1'  longitude. 

Note — Work  out  relationships. 

3.  Why  do  we  estimate  time  from  Greenwich? 

4.  Define  International  Date  line. 

5.  Define  Standard  Time  and  explain  time  belts. 

6.  Examples. 

A.     What  difference  in  time  exists  between  London  and  New 
York  if  longitude  of  London  is  0°  5'  48"  w.  and  that  of  New  York 
is  74°  0'  3"  w.? 

74°       0'       3"     Longitude  of  New  York 
0°       5'      48"     Longitude  of  London 


73°     54'     15"     Difference  in  longitude 

73°  longitude  corresponds  to  4  hrs.  52  min.  time 

54'   longitude  corresponds  to  3  min.  36  sec.  time 

15"  longitude  corresponds  to  1  sec.  time 


4  hrs.  55  min.  37  sec.       time 

B.     If  Boston  is  in  longitude  71°  3'  50"  w.  and  San  Francisco 
is  longitude  122°  25'  42"  w.  what  is  the  difference  in  time? 

122°     25'     42"    Longitude  of  San  Francisco 
71°       3'     50"    Longitude  of  Boston 


51°     21'     52"     Difference  in  longitude 

51°  longitude  corresponds  to  3  hrs.  24  min.  time 

21'   longitude  corresponds  to  1  min.  24  sec.  time 

52"  longitude  corresponds  to  3.46%  sec.       time 

3  hrs.  25  min.  27.46%  sec.        time 

C.    If  two  places  h^ve  2y2  hours  difference  in  time  what  is 
the  difference  in  longitude? 

2i/£  X  15°  =  371/6°  longitude. 


PRACTICAL  MEASUREMENTS 
LENGTH 


1.  Common  units. 

Inch. 

Foot. 

Yard. 

Rod. 

Mile. 

2.  Discuss  measuring  devices. 

Foot  rule. 
Carpenters  rule. 
Yard  stick. 
50  Foot  tape. 
100  Foot  tape. 


3.     Work  out  relationships  through  actual  measurements. 


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SURFACE 

1.  Common  units. 

Square  inch. 
Square  foot. 
Square  yard. 
Square  rod. 
Square  mile. 
Acre. 

2.  Study  these  units  and  \vork  out  table  of  relationships  as  far  as 
possible.     Use  table  for  reference  when  needed. 

3.  Develop  rule  for  finding  area. 

4.  Special  applications  of  square  measure. 

Surface  area  in  inches,  feet,  yards,  rods,  etc. 

Areas  of  floors. 

Areas  of  walls. 

Areas  of  walks. 

Areas  of  roofs. 

Etc. 


IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 

VOLUME 

1.  Common  units. 

Cubic  inch. 
Cubic  foot. 
Cubic  yard. 
Etc. 

2.  Study  these  units  by  making  models,  drawing  pictures,   etc. 
Work  out  relationships  as  far  as  possible. 

3.  Develop  rule  for  finding  volume  of  rectangular  solids. 

4.  Special  applications  of  cubic  measure. 

Volumes^  rooms.  *-.4 

Volume^of  bins 
Volume£bf  boxes. 
Etc. 


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SPECIAL  CASES 

1.  Triangle — Area=1/^   base  X  Alt. 

Circle — Circumference=2f  3 . 1416l^rea%=3 . 1416R 
Trapezoid — Area=i,£  (sum  of  bases)  X  Alt. 
Prism — Lateral  area — Perimeter  of  base  X  Alt. 

Volume=Base  X  Alt. 
Cylinder — Lateral  area=Perimeter  of 

Volume==Base  X  Alt. 
Pyramid — Lateral  area=Perimeter  of  base X  slant  height. 

Volume-=i &  Base X Alt. 
Cone — Lateral  area=Perimeter  of  base  X  slant  height. 

Volume=44  Base  X  Alt. 
Sphere— Area>#3 . 1416jf*Vomme=%  3 . 1416*  * 

2.  Work  out  areas  and  volumes  using  above  formulae. 

3.  Special  units. 

Board  foot. 

Cord. 

Square  (100  sq.  feet). 

Etc. 

4.  All  work  in  mensuration  should  be  presented  through  the  use 
of  objects,  drawings,  tables  for  reference  and  actual  measurement.  A 
text  book  should  be  used  as  a  guide. 


iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmitiii 

THE    EQUATION 

1.  Definition. 

2.  The  four  laws. 

(a)  When  equals  are  multiplied  by  equals  the  results  are 
equal. 

(b)  When   equals   are   divided   by   equals   the   results   are 
equal. 

(c)  When  equals  are  added  to  equals  the  results  are  equal. 

(d)  When   equals   are   taken   from    equals  the  results   are 
equal. 

3.  The   solution  of  equations  when  one  unknown  part  is  to  be 
found. 

4.  The  short  method  of  solution  by  transposition. 

5.  Examples. 

(a)  i£x=3 

x=6  Multiplying  by  2. 

(b)  4x=12 

x=3  Dividing  by  4. 

(c)  x— 2=5 

x=7  Adding  2. 

(d)  x+3=7 

x=4  Subtracting  3. 

(e;     2x— 3=x+5 
2x— x= 

x=8 


491083 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


